Stanford University - ON A FREE BOUNDARY …sg136mg1639/...For a more detailed introduction to...

ON A FREE BOUNDARY PROBLEM FOR EMBEDDED

MINIMAL SURFACES AND INSTABILITY THEOREMS FOR

MANIFOLDS WITH POSITIVE ISOTROPIC CURVATURE

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MATHEMATICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Man Chun Li

August 2011

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/sg136mg1639

© 2011 by Man Chun Li. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/sg136mg1639

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Richard Schoen, Primary Adviser


Leon Simon


Brian White

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

In this thesis, we describe a min-max construction of embedded minimal surfaces

satisfying the free boundary condition in any compact 3-manifolds with boundary.

We also prove the instability of minimal surfaces of certain conformal type in 4-

manifolds with positive isotropic curvature.

Given a compact 3-manifold M with boundary ∂M , consider the problem of find-

ing an embedded minimal surface Σ which meets ∂M orthogonally along ∂Σ. These

surfaces are critical points to the area functional with respect to variations preserving

∂M . We will use a min-max construction to construct such a free boundary solution

and prove the regularity of such solution up to the free boundary. An interesting point

is that no convexity assumption on ∂M is required. We also discuss some geometric

properties, genus bounds for example, for these free boundary solutions.

Just as positive sectional curvature tends to make geodesics unstable, positive

isotropic curvature tends to make minimal surfaces unstable. In the second part of

this thesis, we prove a similar instability result in dimension 4. Given a compact 4-

manifold M with positive isotropic curvature, we show that any complete immersed

minimal surface Σ in M which is uniformly conformally equivalent to the complex

plane is unstable. The same conclusion holds in higher dimensions as well if we

assume that the manifold has uniformly positive complex sectional curvature. The

proof uses the Hormander’s weighted L2 method and the stability inequality to derive

a contradiction.

iv

Acknowledgments

I am deeply grateful to my adviser, Professor Richard Schoen, for suggesting this the-

sis topic and all his continuous support, understanding and encouragement through-

out the progress of this work. His insight, passion and persistence in Mathematics

have tremendous positive influence on my education as a mathematician. I would like

to thank him sincerely.

I would also like to express my gratitude to Professor Leon Simon and Professor

Brian White, for introducing me to the field of geometric measure theory and all the

useful conversations. I thank the department of Mathematics in CUHK, especially

Professor Tom Wan, Professor Conan Leung and Professor Thomas Au, for nurturing

my interest in geometry as an undergraduate. I would also like to thank the Math-

ematics department at Stanford, Professor Simon Brendle, Professor Rafe Mazzeo,

Professor Andras Vasy and Professor Yakov Eliashberg, for all the beautiful mathe-

matics that they taught me. I am also thankful to my fellow classmates and friends,

Jesse Gell-Redman, Man Chuen Cheng, Xin Zhou, Frederick Fong for many fruitful

discussions.

I wish to acknowledge the Sir Edward Youde Memorial Fund for their financial

support during my first three years of graduate school.

I also thank my family for their love and support, especially my mother, who con-

sistently taught her children the importance of education, and respected my decisions

on all aspects.

Finally, I especially would like to thank my fiancee Jane Dai, without whom I could

not have finished this thesis so smoothly. Jane brings me happiness and inspiration,

supports me at difficult times, and pushes me forward when I lack the motive for

v

improvement. We dream of exploring the world of science together, with the greatest

appreciation of its beauty and mystery.

vi

Dedicated to Jane,

my dearest

vii

Contents

Abstract iv

Acknowledgments v

1 Introduction 1

1.1 Existence of minimal surfaces . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Plateau’s problem . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Area-minimzing surfaces . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Min-max construction of minimal surfaces . . . . . . . . . . . 5

1.1.4 Minimal surfaces with free boundaries . . . . . . . . . . . . . 7

1.2 Manifolds with positive isotropic curvature . . . . . . . . . . . . . . . 10

1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Topology of manifolds with positive isotropic curvature . . . . 12

1.2.3 Second variation of energy and isotropic curvature . . . . . . . 13

2 Minimal Surfaces with Free Boundary 16

2.1 Definitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Isotopies and vector fields . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Varifolds and restrictions . . . . . . . . . . . . . . . . . . . . . 19

2.1.3 First variation formula and its consequences . . . . . . . . . . 21

2.2 The min-max construction . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Sweep-outs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 The min-max construction . . . . . . . . . . . . . . . . . . . . 25

2.2.3 The main result and a technical lemma . . . . . . . . . . . . . 27

viii

2.3 Existence of stationary varifolds . . . . . . . . . . . . . . . . . . . . . 32

2.4 Existence of almost minimizing sequence . . . . . . . . . . . . . . . . 37

2.5 A minimization problem with partially free boundary . . . . . . . . . 43

2.5.1 Local γ-reductions . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5.2 Minimizing sequence of genus 0 surfaces . . . . . . . . . . . . 57

2.5.3 Convergence of the minimizing sequence . . . . . . . . . . . . 58

2.6 Regularity of almost minimizing varifolds . . . . . . . . . . . . . . . . 61

2.7 Genus bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 Minimal surfaces in PIC manifolds 70

3.1 Definitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 A vanishing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3 Hormander’s weighted L2 method . . . . . . . . . . . . . . . . . . . . 73

3.4 The main theorem and its proof . . . . . . . . . . . . . . . . . . . . . 79

3.5 A density lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Bibliography 84

ix

Chapter 1

Introduction

In this chapter, we survey some old and new results about minimal surfaces . The

field of minimal surfaces has its origin in the mid eighteenth century with the work

of Euler and Lagrange on calculus of variations. The study of minimal surfaces is

very interesting as it lies at the intersection of nonlinear elliptic partial differential

equations, geometry, topology and general relativity. This is why minimal surfaces

has remained a vibrant area of research in mathematics since the eighteenth century.

Minimal surfaces arises naturally in physical situations as well. In the nineteenth

century, Belgian physicist Joseph Plateau studied extensively the physics of soap films

and soap bubbles. Since then it has been known that minimal surfaces are the right

mathematical model for soap films, which represent a state of minimum energy while

covering the least possible amount of area. As many physical laws (the law of least

action, for example) or equations come from minimizing or finding critical points of

certain functionals (which physicists call action), minimal surfaces and equations like

the minimal surface equation have served as mathematical models for many physical

problems.

To begin, let us define minimal surfaces in mathematical terms. Let Σ ⊂ Rn be

a smooth surface (possibly with boundary) and C∞0 (NΣ) be the space of all smooth

compactly supported normal vector fields on Σ. Given Φ ∈ C∞0 (NΣ), consider the

1

CHAPTER 1. INTRODUCTION 2

one-parameter family of surfaces

Σt,Φ = {x+ tΦ(x)|x ∈ Σ},

and the first variation formula for the area functional:

d

dt

∣∣∣∣t=0

Area(Σt,Φ) = −∫

Σ

〈Φ, H〉dvol, (1.1)

where H is the mean curvature vector of Σ in Rn. (If Σ is noncompact or has

boundary, we require Φ to be supported in a compact set disjoint from ∂Σ.) Thus, Σ

is minimal if and only if it is a critical point for the area functional, which is equivalent

to the condition that the mean curvature vector H vanishes identically.

In the first part of this chapter, we will give a survey on the existence of minimal

surfaces, with various kinds of boundary conditions. The literature on this subject

is vast and we are by no means trying to be exhaustive here. In the second part,

we will see some applications of minimal surfaces in geometry through the relation-

ship between stability of minimal surfaces and curvatures of the ambient manifolds.

This subtle relationship has been very useful in the study of manifolds with positive

curvatures, in particular for manifolds with positive scalar curvature, which gives a

number of physical applications in general relativity as well.

For a more detailed introduction to minimal surfaces, one refers to the recent book

by T. Colding and W. Minicozzi [12].

1.1 Existence of minimal surfaces

1.1.1 Plateau’s problem

The very first existence result is the Plateau’s problem: given a simple closed curve Γ

in R3, can we find a minimal surface spanning the boundary Γ? There are various solu-

tions to this problem depending on the exact definition of a surface (parametrized disk,

integral current, Z2-current, or rectifiable varifold). The question for parametrized


disk was first answered affirmatively by T. Rado [46] and J. Douglas [14] indepen-

dently. The generalization to Riemannian manifolds is due to C. B. Morrey [41].

Theorem 1.1.1 (Douglas [14], Rado [46]). Let Γ ⊂ R3 be a piecewise C1 closed

Jordan curve. Then there exists a piecewise C1 map u : D ⊂ R2 → R3, where D

is the closed unit disk, and u maps ∂D monotonically onto Γ, such that the image

minimizes area among all disks with boundary Γ.

The solution u to the Plateau’s problem above can easily be seen to be a branched

conformal immersion. R. Osserman [43] proved that u does not have true interior

branch points; subsequently, R. Gulliver [25] and H. W. Alt [3] showed that u cannot

have false branch points either.

Furthermore, the solution u is as smooth as the boundary curve, even up to

the boundary. A very general version of this boundary regularity was proved by S.

Hildebrand [28]; for the case of surfaces in R3, recall the following result of J. Nitsche

[42]:

Theorem 1.1.2 (Nitsche [42]). If Γ is a regular Jordan curve of class Ck,α where

k ≥ 1 and 0 < α < 1, then a solution u of the Plateau’s problem is Ck,α on all of D.

The optimal boundary regularity theorem in higher dimensions was proven by R.

Hardt and L. Simon [27].

However, such a solution is only an immersed disk but not always embedded, i.e.

there may be self-intersections. If one wants to find an embedded minimal disk,

then we need some convexity assumption on the boundary curve Γ to guarantee the

existence of embedded solutions. F. Almgren and L. Simon [2], W. Meeks and S. T.

Yau [38] obtained some positive results along this direction using completely different

approaches. They showed that a solution to the embedded Plateau’s problem exists

if the boundary curve Γ lies on the boundary of a bounded uniformly convex open

set in R3. We will say more about this in the next section.

1.1.2 Area-minimzing surfaces

Perhaps the most natural way to construct minimal surfaces is to look for ones which

minimize area, for example, with fixed boundary, or in a homotopy class, or in a


homology class, etc. This has the advantage that often it is possible to show that the

resulting surface is embedded. We mention a few results along these lines.

The first embeddedness result, due to W. Meeks and S.T. Yau [38], shows that if

the boundary curve is embedded and lies on the boundary of a smooth mean convex

set (and it is null-homotopic in this set hence it bounds at least one parametrized

disk), then it bounds an embedded least area disk.

Theorem 1.1.3 (Meeks-Yau [38]). Let M3 be a compact Riemannian 3-manifold

whose boundary is mean convex and let γ be a simple closed curve in ∂M which is

null-homotopic in M , then γ is bounded by a least area disk and any such least area

disk is properly embedded.

Note that some restrictions on the boundary curve γ is certainly necessary. For

instance, if the boundary curve was knotted (for example, the trefoil), then it could

not be spanned by any embedded disk (minimal or otherwise). Prior to the work of

W. Meeks and S. T. Yau, embeddedness was known for extremal boundary curves

in R3 with small total curvature by the work of R. Gulliver and J. Spruck [24].

Subsequently, F. Almgren and L. Simon [2], F. Tomi and A. Tromba [54] proved

the existence of some embedded solution for extremal boundary curves in R3 (but

not necessarily the Douglas-Rado solution). Recently, T. Ekholm, B. White and D.

Wienholtz [15] proved that minimal surfaces whose boundary has total curvature less

than 4π must be embedded.

If we instead fix a homotopy class of maps, then the two fundamental existence

results are due to J. Sacks and K. Uhlenbeck [47], R. Schoen and S. T. Yau [48] (with

embeddedness proved by W. Meeks and S. T. Yau [37], M. Freedman, J. Hass and P.

Scott [19] respectively):

Theorem 1.1.4 (Sacks-Uhlenbeck [47], Meeks-Yau [37]). Given M3, there exists

conformal (stable) minimal immersions u1, . . . , um : S2 → M which generate π2(M)

as a Z[π1(M)]-module. Furthermore,

• If u : S2 →M and [u]π2 6= 0, then Area(u) ≥ mini Area(ui).

• Each ui is either an embedding or a 2-1 map onto an embedded 2-sided RP 2.


Theorem 1.1.5 (Schoen-Yau [48], Freedman-Hass-Scott [19]). If Σ2 is a closed sur-

face with genus g > 0 and i0 : Σ → M3 is an embedding which induces an injective

map on π1, then there is a least area embedding with the same action on π1.

In [35], W. Meeks, L. Simon and S. T. Yau find an embedded sphere minimizing

area in any isotopy class in a closed 3-manifold using geometric measure theoretic

techniques.

1.1.3 Min-max construction of minimal surfaces

Variational arguments can also be used to construct higher index (i.e., non-minimizing)

minimal surfaces using the topology of the space of surfaces. There are two basic ap-

proaches:

• Applying Morse theory to the energy functional on the space of maps from a

fixed surface Σ to M .

• Doing a min-max argument over families of (topologically non-trivial) sweep-

outs of M .

The first approach has the advantage that the topological type of the minimal surface

is easily fixed; however, the second approach has been more successful at producing

embedded minimal surfaces. We will highlight a few key results below but refer to

Colding-De Lellis [9] for a thorough treatment.

Unfortunately, one cannot directly apply Morse theory to the energy functional

on the space of maps from a fixed surface because of the lack of compactness (the

Palais-Smale Condition C does not hold). To get around this difficulty, J. Sacks

and K. Uhlenbeck [47] introduced a family of perturbed energy functionals which do

satisfy Condition C and the obtained minimal surfaces as limits of critical points for

the perturbed problems:

Theorem 1.1.6 (Sacks-Uhlenbeck [47]). If πk(M) 6= 0 for some k > 1, then there

exists a branched immersed minimal 2-sphere in M for any metric.


This was sharpened somewhat by M. Micallef and J. Moore [39] (showing that the

index of the minimal sphere was at most k−2), who used it to prove a generalization

of the sphere theorem. See Fraser [17] for a generalization to a free boundary problem.

The basic idea of constructing minimal surfaces via min-max arguments and

sweep-outs goes back to Birkhoff, who developed it to construct simple closed geodesics

on spheres. In particular, when M is a topological 2-sphere, we can find a one-

parameter family of curves starting and ending at point curves so that the induced

map F : S2 → S2 has non-zero degree. The min-max argument produces a non-trivial

closed geodesic of length less than or equal to the longest curve in the initial one-

parameter family. A curve shortening argument gives that the geodesic obtained in

this way is simple.

Using the topology of the space of integral currents, F. Almgren [33] introduced the

concept of varifolds and a min-max construction to prove the existence of stationary

varifolds in any dimensions and codimensions. Later, J. Pitts [44] elaborated on the

idea of Almgren and proved regularity in the codimension one case. Hence, he showed

that every closed Riemannian 3-manifold has an embedded smooth minimal surface

(his argument was for dimension up to seven), but he did not estimate the genus of

the resulting surface. Finally, F. Smith [52] in his PhD dissertation proved

Theorem 1.1.7 (Smith [52]). Every metric on a topological 3-sphere M admits an

embedded minimal 2-sphere.

The main new contribution of Smith was to control the topological type of the

resulting minimal surface while keeping it embedded. Later, Pitts and Rubinstein

[45] announced a more general genus bound for minimal surfaces arising from these

min-max constructions but their proof was never published. A slightly weaker genus

bound was proved recently by De Lellis and Pellandini in [13]. In [45], they also

claim without proof an index bound for these min-max surfaces. Very recently, some

positive results on index bound are obtained by F. Marques and A. Neves.

We also want to mention the recent work by T. Colding and W. Minicozzi [11]

[10] which uses these min-max minimal surfaces to define the concept of width of a

manifold. They proved an estimate on the rate of change of width which implies the


finite extinction time for Ricci flow and mean curvature flow.

1.1.4 Minimal surfaces with free boundaries

Consider a compact 3-manifold M with non-empty boundary ∂M , we will prove the

existence of a non-trivial compact embedded minimal surface Σ with free boundary

lying on ∂M . In case M is smooth up to the boundary, we see that Σ is also smooth

up to the boundary.

Theorem 1.1.8. Any compact Riemannian 3-manifold M with non-empty boundary

∂M admits a non-trivial compact embedded minimal surface Σ with free boundary ∂Σ

lying on ∂M . In other words, Σ meets ∂M orthogonally along ∂Σ.

The main difficulty in the proof of Theorem 1.1.8 comes from the fact that we

do not have any convexity assumptions on the boundary ∂M . Unlike in many con-

structions of minimal surfaces, we no longer have “barriers” to prevent our minimal

surfaces from touching the boundary. For example, M. Gruter and J. Jost [21] [30] [31]

[32] looked into this free boundary problem for embedded minimal disks, assuming

that M is diffeomorphic to a 3-ball (and some convexity assumptions on ∂M).

We overcome this difficulty by first embedding M into a bigger 3-manifold M ,

then we allow isotopies to move points out of M . However, we will only count the

area of the isotopic surfaces inside the compact set M . Since we are allowing more

admissible deformations, we need to be careful about the meaning of a stationary

surface and the definition ofalmost minimizing property which is crucial in proving

regularity.

Given a smooth embedded surface Σ, possibly with boundary, in M and a 1-

parameter family of diffeomorphisms {ϕt}t∈(−ε,ε) of M generated by a smooth vector

field X in M , consider the family {ϕt(Σ)}t∈(−ε,ε) of smooth embedded surfaces in M ,

the area changes smoothly in t so it makes sense to compute its derivative at t = 0.

This gives the classical first variation formula for area. A surface (or more generally

a varifold) is said to be stationary if the first variation vanishes for all vector fields.

However, if we only count the area inside the compact subset M ⊂ M , the area of

ϕt(Σ) in M no longer varies smoothly in t (in fact it is not even continuous in general).


Therefore, the first variation formula for area in M does not make sense in this case.

To get around this technical difficulty, we restrict to a smaller class of variations in M ,

namely, we only look at variations which preserve the compact set M . Infinitesimally,

this means we are only looking at vector fields which are tangential to ∂M at the

boundary. For these vector fields, the first variation formula for area in M makes

perfect sense, and we say that a varifold is tangentially stationary if the first variation

vanishes for all tangential vector fields.

However, tangentially stationary varifolds can be quite far from being a free bound-

ary minimal surface. Let’s consider the following example (see figure 1 below). Sup-

pose M is the unit ball in R3 minus the ball of radius 1/4 centered at (0, 0, 1/4). The

equatorial unit disk is easily seen to be tangentially stationary, yet it is not a free

boundary surface because there is an interior point, namely the origin, touching the

boundary ∂M tangentially, not orthogonally. In this example, the boundary ∂M is

disconnected but one can easily modify the above example so that M is diffeomorphic

to a ball. This phenomenon occurs because the boundary of M is not convex at the

origin. On the other hand, M. Gruter and J. Jost [21] [32] constructed embedded

minimal disks with free boundary when M is diffeomorphic to the three dimensional

ball, assuming ∂M is convex (or mean convex). The convexity prevents the interior

of the minimal disk from touching the boundary by maximum principle, hence it was

sufficient to consider variations which preserve the boundary ∂M .

Knowing a surface is tangentially stationary is not enough to prove regularity, the

key ingredient in proving regularity is the almost minimizing property, which was first

introduced by J. Pitts in [44]. However, it is not difficult to see that the example

above is almost minimizing with respect to variations preserving the boundary ∂M .

Therefore, to get a genuine free boundary solution we need a way to make the interior

points of the surface detached from the boundary ∂M . One way to achieve this is

to allow our variations to move points out of the boundary ∂M and then restrict

the perturbed surface back to M . In this way we are able to decrease the area of

the surface in M without having to increase it first by a certain amount. Hence, the

surface is not almost minimizing with respect to “outward” variations.

In summary, we restrict to tangential variations for stationarity to get continuity


and compactness results while we allow more variations for the almost minimizing

property to prove regularity at the free boundaries.

We should mention that there is another min-max construction for free boundary

minimal surfaces, sometimes called the mapping problem. For these constructions,

one fixes the topological type of the minimal surface and look for critical points

for the Dirichlet integral among maps from the surface satisfying the free boundary

condition. The free boundary problem was first considered by R. Courant who proved

the existence of a branched minimal immersion of the disk into the solid torus. In a

series of papers, W. Meeks and S. T. Yau [36] [37] [38] proved the geometric Dehn’s

lemma, loop theorem and sphere theorem in 3-manifold theory. They solved a free

boundary problem by minimizing the area among all maps from the unit disk into

the manifold which map the unit circle to a homotopically non-trivial closed curve in

the boundary of the manifold.

Theorem 1.1.9 (Meeks-Yau [37]). Let M be a compact Riemannian manifold with

mean convex boundary ∂M . Let F be the family of all smooth maps f from the unit

disk D into M so that f : ∂D → ∂M represents a non-trivial element in π1(∂M).

Then there exists a conformal harmonic map in F which has minimal area compared

with all the other elements in F. Furthermore, f maps the normal vector of ∂D to a

non-zero normal vector of ∂M .

In case the supporting surface ∂M is a sphere, M. Struwe [53] showed that for

any closed surface S ⊂ R3 diffeomorphic to S2 the existence of a non-trivial mini-

mal surface intersecting S orthogonally along its boundary. However, the minimal

surface may not lie in the compact set bounded by S unless we have some convexity

assumption on S. R. Ye [56] considered the free boundary problem with prescribed

free homotopy class on the boundary curve, he proved that if a free homotopy class

satisfies the “Douglas condition”, then there exists an area-minimizing surface with

free boundary. A. Fraser [17] studied the free boundary problem for minimal disks

along the ideas of J. Sacks and K. Uhlenbeck [47] and obtained some existence results

with index bounds using Morse theoretic techniques . The boundary regularity for

the free boundary problem was discussed in Grrter-Hildebrandt-Nitsche [22]. One


gets better control of the topological type of the minimal surface produced, however,

they are only immersed surfaces, in general, and may contain isolated branch points.

Therefore, the minimal surfaces constructed in chapter 2 are more geometric because

they are always embedded. The advantage of geometric measure theoretic method is

that it handles the regularity and embeddedness simultaneously.

1.2 Manifolds with positive isotropic curvature

1.2.1 Definitions

We refer the reader to J. Cheeger and D. Ebin’s book [6] for basic facts and notions

in Riemannian geometry.

Let Mn be an n-dimensional Riemannian manifold. Consider the complexified

tangent bundle TCM = TM ⊗RC equipped with the Hermitian extension 〈·, ·〉 of the

inner product on TM , the curvature tensor extends to complex vectors by linearity,

and the complex sectional curvature of a complex two-dimensional subspace π of TCp M

at some point p ∈M is defined by KC(π) = 〈R(v, w)w, v〉, where {v, w} is any unitary

basis for π.

Definition 1.2.1. A Riemannian manifold Mn has positive complex sectional curva-

ture if KC(π) > 0 for every complex two-dimensional subspace π in TCp M at every

p ∈ M . A Riemannian manifold Mn has uniform positive complex sectional curva-

ture if there exists a constant κ > 0 such that KC(π) ≥ κ > 0 for every complex

two-dimensional subspace π in TCp M at every p ∈M .

Remark 1.2.2. It is clear that having uniformly positive complex sectional curvature

implies having uniformly positive sectional curvature. (One simply considers all π

which comes from the complexification of a real two-dimensional subspace in TpM .)

Therefore, by Bonnet-Myers theorem, M is automatically compact if it is complete.

Using Ricci flow techniques, S. Brendle and R. Schoen [5] proved that any manifold

M with uniformly positive complex sectional curvature is diffeomorphic to a spherical

space form. In particular, when M is simply-connected, M is diffeomorphic to the n-

dimensional sphere. In fact, S. Brendle and R. Schoen [5] showed that the condition


of positive complex sectional curvature is preserved under the Ricci flow, and any

manifold equipped with such a metric evolves under the normalized Ricci flow to a

spherical space-form. The optimal convergence result so far is obtained by S. Brendle

in [4], where he proved that any compact manifold M such that M × R has positive

isotropic curvature would converge to a spherical space form under the normalized

Ricci flow.

As a result, having positive complex sectional curvature is a rather restrictive

condition. There is a related positivity condition which allows more flexibility called

positive isotropic curvature (PIC). Instead of extending in a Hermitian way, one can

choose to extend the metric on TM to a C-bilinear form (·, ·) on TCM . We say that

a vector v ∈ TCp M is isotropic if (v, v) = 0. A subspace V ⊂ TC

p is isotropic if every

v ∈ V is isotropic.

Definition 1.2.3. A Riemannian manifold Mn, n ≥ 4, has positive isotropic cur-

vature (PIC) if KC(π) > 0 for every isotropic complex two-dimensional subspace

π ⊂ TCp M at every p ∈M .

A Riemannian manifold Mn, n ≥ 4, has uniformly positive isotropic curvature

(uniformly PIC) if there exists a positive constant κ > 0 such that KC(π) ≥ κ > 0

for every isotropic complex two-dimensional subspace π ⊂ TCp M at every p ∈M .

Remark 1.2.4. This condition is clearly weaker than the previous one because it only

requires positivity of complex sectional curvature among isotropic 2-planes. Moreover,

the condition is vacuous when n ≤ 3 (see [39]), so we restrict ourselves to n ≥ 4.

The notion of positive isotropic curvature can be formulated purely in real terms

as follows: (M, g) has positive isotropic curvature if and only if

R1313 +R1414 +R2323 +R2424 − 2R1234 > 0 (1.2)

for all orthonormal 4-frames (e1, e2, e3, e4), where Rijkl = R(ei, ej, ek, el) is the Rie-

mann curvature tensor and Rijij is the sectional curvature of the two-plane spanned

by ei and ej.

Typical examples of manifolds with positive isotropic curvature includes manifolds


with positive curvature operator (for example, the standard round sphere Sn), man-

ifolds with pointwise strictly quarter-pinched sectional curvatures, and the product

metric on Sn−1 × S1.

1.2.2 Topology of manifolds with positive isotropic curvature

In the paper [5], S. Brendle and R. Schoen showed that uniformly PIC is also a

condition preserved under the Ricci flow. However, it is not true that any manifold

equipped with such a metric would converge to a spherical space-form under the

normalized Ricci flow. For example, the product manifold Sn−1 × S1 is PIC but it

has no metric with constant positive sectional curvature. (The universal cover of

Sn−1 × S1 is Sn−1 × R, which is not Sn.)

On the other hand, there is a “sphere theorem” for compact manifolds with PIC.

Namely, if Mn, n ≥ 4, is a compact manifold with PIC, and M is simply-connected,

then Mn is homeomorphic to Sn. This sphere theorem, which generalizes the classical

sphere theorem since any 1/4-pinched manifold is PIC, was proved by M. Micallef

and J. Moore in [39], where the notion of PIC was first defined. In other words,

all simply-connected PIC manifolds are topologically “trivial”. This, of course, raises

the natural question: what are the topological obstructions on the fundamental group

π1M for a compact manifold M to admit a metric with PIC? This lead to a conjecture

due to Gromov:

Conjecture 1.2.5 (Gromov). Let (Mn, g) be a compact Riemannian manifold with

positive isotropic curvature. Then, the fundamental group π1(M) is virtually free,

i.e. π1(M) contains a free subgroup with finite index.

Along this direction, A. Fraser [18] proved that for n ≥ 5, π1M cannot contain a

subgroup isomorphic to Z⊕Z, the fundamental group of a torus. Her proof uses the

existence theory of stable minimal surfaces by R. Schoen and S.T. Yau [48], and the

Riemann-Roch theorem to construct almost holomorphic variations which, in turn,

contradicts stability. A few years later, S. Brendle and R. Schoen [5] proved that

the same result holds for n = 4. On the other hand, M. Micallef and M. Wang [40]

proved that if (Mn1 , g1) and (Mn

2 , g2) are manifolds with positive isotropic curvature,


then M1]M2 also admits a metric with positive isotropic curvature. In particular,

]ki=1(S1 × Sn−1) admits a metric with positive isotropic curvature for any positive

integer k. Therefore, the fundamental group of a manifold with positive isotropic

curvature can be quite large.

An important conjecture towards the classification of compact manifolds with

positive isotropic curvature is due to Schoen.

Conjecture 1.2.6 (Schoen). Let (Mn, g), n ≥ 4 be a compact closed Riemannian

n-manifold with positive isotropic curvature. Then, there exists a finite covering M

such that M is diffeomorphic to Sn](Sn−1 × S1)] · · · ](Sn−1 × S1).

In dimension four, R. Hamilton [26] outlined a way of proving Schoen’s conjecture

using Ricci flow with surgery and the details are carried out by B. L. Chen and X. P.

Zhu in [7]. On the other hand, S. Gadgil and H. Seshadri [20] proved that Gromov’s

conjecture implies part of Schoen’s conjecture: if (M, g) has PIC and free fundamental

group, then M is homeomorphic to a connected sum of copies of Sn−1 × S1.

For even-dimensional manifolds M , M. Micallef and M. Y. Wang [40], W. Seaman

[50] independently proved that PIC implies the vanishing of the second Betti number

of M . In fact, they showed that the curvature term in the Bochner formula for

harmonic 2-forms can be expressed in terms of the isotropic curvatures. Hodge theory

then gives the desired result.

1.2.3 Second variation of energy and isotropic curvature

It has been a central theme in Riemannian geometry to study how the curvatures of

a manifold affect its topology. For 2-dimensional surfaces, the Gauss-Bonnet theorem

plays a key role. In particular, it implies that any oriented closed surface with positive

sectional curvature is diffeomorphic to the 2-sphere. In higher dimensions, Synge

theorem says that any closed even-dimensional oriented Riemannian manifold M2n

with positive sectional curvature is simply connected. The key idea in the proof is

that there exists no stable closed geodesic σ in such manifold. To see this, recall

the second variation formula for lengths of a 1-parameter family of closed curves


{σt}t∈(−ε,ε):d2L(σt)

dt2

∣∣∣∣t=0

=

∫σ

[‖∇σ′X‖2 − 〈R(σ′, X)σ′, X〉] ds

where σ = σ0 is a closed geodesic in M , σ′ = ∂σ∂s

is the tangent vector along σ

with s being the arc-length parameter, X = ∂σt∂t

∣∣t=0

is a variation field along σ and

R(X, Y )Z = ∇Y∇XZ −∇X∇YZ +∇[X,Y ]Z is the Riemann curvature tensor for M .

Synge observed that if M is even-dimensional and oriented, then there is a non-zero

parallel vector field X (i.e. ∇σ′X = 0), hence positivity of sectional curvature implies

thatd2L(σt)

dt2

∣∣∣∣t=0

< 0.

Consequently, any closed geodesic in an oriented Riemannian manifold M2n with

positive sectional curvature is unstable. Therefore, M has to be simply connected.

Otherwise, one could minimize the length in any non-trivial free homotopy class in

π1(M) to get a stable closed geodesic, which is a contradiction.

The idea of Synge can be extended in various directions. One can consider smooth

maps φ : S2 → M and the energy E of such maps. The critical points of E are

conformal branched minimal immersions of S2 into M . The Hessian of E is a bilinear

form on the space of sections of the pullback bundle φ∗TM . As before, we can

complexify the bundle φ∗TM to get V = φ∗TM ⊗ C and consider the complex

linear extension of the Hessian to the space of sections of V . The metric on φ∗TM

can be extended as a Hermitian metric 〈·, ·〉 or a C-bilinear form (·, ·) on V and

the connection can be extended in a C-linear fashion. Moreover, V can be given a

holomorphic structure so that a section s is holomorphic if and only if

∂s = ∇ ∂∂zs = 0.

We then have

I(s, s) =

∫S2

(‖∂s‖2 − 〈R(s ∧ ∂f

∂z), s ∧ ∂f

∂z〉)dx dy, (1.3)

where z = x+√−1y is any local holomorphic coordinate, ∂f

∂z= f∗(

∂∂z

), and R is the


complexified (extended C-linearly) curvature operator on ∧2TM ⊗ C.

The fact that f is conformal implies that

(∂f

∂z,∂f

∂z) = 0,

i.e. ∂f∂z

is an isotropic section. If s is a holomorphic section such that s and ∂f∂z

span an isotropic 2-plane, then the second term in the integral represents an isotropic

curvature. Hence, positive isotropic curvature tends to make minimal surfaces unsta-

ble. Using this idea, M. Micallef and J. Moore [39] proved an index lower bound for

harmonic 2-spheres in manifolds with positive isotropic curvature.

Theorem 1.2.7 (Micallef-Moore [39]). Let (Mn, g), n ≥ 4, be a compact mani-

fold with positive isotropic curvature. Then, any nonconstant conformal harmonic

2-sphere in M has index at least n−32

.

Combining with a Morse theoretic argument on the existence of harmonic 2-

spheres with low index, if M is simply connected, they concluded that M is a homo-

topy sphere and hence is a topological sphere by the generalized Poincare conjecture.

In case M is not simply connected, A. Fraser [18] proved that π1(M) cannot con-

tained any subgroup isomorphic to Z ⊕ Z. The proof based on the fact that any

immersed minimal torus in M has a finite covering which is unstable. In chapter 3,

we will prove that, in dimension four, any complete minimal surfaces in M which are

uniformly conformally equivalent to C must be unstable. The proof goes by construct-

ing holomorphic sections (half-parallel sections) with slow growth using Hormander’s

weighted L2-method and then apply a weighted second variation argument. It is

conceivable that such a result could have topological implications similar to the ones

obtained by A. Fraser.

Chapter 2

Existence of embedded minimal

surfaces with free boundary

In this chapter, we will prove Main Theorem A below concerning the existence of

embedded minimal surfaces with free boundary.

Theorem 2.0.8 (Main Theorem A). Let (M, g) be a compact C∞ Riemannian 3-

manifold with boundary ∂M . There exists an embedded C∞ minimal surface Σ ⊂M with (possibly empty) boundary ∂Σ such that Σ ∩ ∂M = ∂Σ and Σ meets ∂M

orthogonally along ∂Σ.

Furthermore, any compact 3-manifold (M3, g) with filling genus h (see Section 2.7

for definition) contains a non-trivial embedded minimal surface Σ with free boundary

such that one of the following holds:

(i) Σ is orientable with genus less than or equal to h.

(ii) Σ is non-orientable with genus less than or equal to 2h+ 1.

An important corollary is the following:

Corollary 2.0.9 (Main Corollary). Any smooth compact domain in R3 contains a

non-trivial embedded minimal surface Σ with non-empty boundary which is a free

boundary solution and such that

16

CHAPTER 2. MINIMAL SURFACES WITH FREE BOUNDARY 17

(i) either Σ is an orientable genus zero surface, i.e. a disk with holes;

(ii) or Σ is a non-orientable genus one surface, i.e. a Mobius band with holes.

Note that the condition Σ ∩ ∂M = ∂Σ implies that Σ \ ∂Σ ⊂ M \ ∂M and

∂Σ ⊂ ∂M . In other words, the interior of Σ is contained in the interior of M , and

the boundary of Σ lies on the boundary of M . Therefore, no interior point of Σ lies

on the boundary of M . This is rather non-trivial since we do not have any convexity

assumption on ∂M . As a result, we cannot simply apply the maximum principle to

rule out such a situation.

In section 2.1, we describe our setup and notations for the free boundary problem

and discuss a few preliminary lemmas which will be used throughout this chapter.

In section 2.2, we explain the min-max construction and prove a technical pertur-

bation lemma about transversality. In section 2.3, we use the natural Birkhoff area-

decreasing process to tighten our minimizing sequence of sweep-outs. This forces

almost-maximal slices to be almost-stationary. In section 2.4, we prove the existence

of almost-minimizing sequences with respect to outward deformations. In section 2.5,

we discuss a minimization problem for partially free boundary and prove a regular-

ity result for limits of minimizing sequences. In section 2.6, we use the results from

section 2.4 and 2.5 to construct replacements, and we show that having sufficiently

many replacements implies that the stationary varifold is in fact smooth. In section

2.7, we will discuss genus and index bounds for these min-max surfaces.

2.1 Definitions and preliminaries

Throughout this chapter, we assume all manifolds and maps are C∞ unless otherwise

stated. If there is a boundary, we assume they are smooth up to the boundary as

well.

Let (M3, g) be a compact Riemannian 3-manifold with non-empty boundary

∂M 6= φ. Suppose M is connected (but ∂M is not necessary connected, i.e. M

could have multiple boundary components.) As discussed in Section 1.1.4, without

loss of generality, we assume that M is isometrically embedded as a compact subset of


a closed Riemannian 3-manifold M . (Note that such an isometric embedding always

exists. For example, we can extend M across the boundary ∂M to get a collar neigh-

borhood which can be made cylindrical near the boundary by a cutoff argument, then

take another copy of this collar neighborhood and glue the two together along the

cylindrical necks.) All surfaces (with or without boundary) are smoothly embedded

in M unless otherwise stated. We will use int(M) to denote the interior of M .

2.1.1 Isotopies and vector fields

We want to describe the class of ambient isotopies in M used in deforming our surfaces.

An isotopy on M is a 1-parameter smooth family of diffeomorphisms of M , say

{ϕs}s∈[0,1], parametrized by the interval [0, 1], where ϕ0 is the identity map of M . (The

smoothness assumption here means that the map ϕ(s, x) = ϕs(x) : [0, 1] × M → M

is smooth.) Let Is denote the space of all isotopies on M . Moreover, we say that an

isotopy {ϕs} ∈ Is is supported in an open set U ⊂ M if ϕs(x) = x for every s ∈ [0, 1]

and x ∈ M \U . As discussed in the introduction, we need to consider isotopies in M

which can move points out of the compact set M ⊂ M , but not into M . Hence, we

define

Isout = {{ϕs} ∈ Is : M ⊂ ϕs(M) for all s ∈ [0, 1]}.

Similarly, we define Isout(U) to be isotopies in Isout which are supported in some

open set U ⊂ M . Furthermore, we are also interested in the situations where the

compact set M is preserved by the isotopy, so we define

Istan = {{ϕs} ∈ Is : M = ϕs(M) for all s ∈ [0, 1]}.

Similarly, Istan(U) consists of those in Istan such which are supported in some open

set U ⊂ M . Notice that Istan(U) ⊂ Isout(U) for any open set U ⊂ M .

One way to generate isotopies is to consider the flow of a vector field. Let

C∞(M, TM) be the vector space of smooth vector fields on M . We will define two

subspaces C∞out(M, TM) and C∞tan(M, TM) of C∞(M, TM) which correspond to the


two classes of isotopies defined above. We define

C∞out(M, TM) = {X ∈ C∞(M, TM) : For any x ∈ ∂M, X(x) · ν ≤ 0,

where ν is the unit inward pointing normal of ∂M}

and

C∞tan(M, TM) = {X ∈ C∞(M, TM) : X(x) ∈ Tx∂M for any x ∈ ∂M}.

Notice again that C∞tan(M, TM) ⊂ C∞out(M, TM). Each X ∈ C∞(M, TM) generates

a unique isotopy {ϕs}s∈[0,1] on M by its flow. (Since M is closed, the isotopy exists

for all s ∈ R.) If X ∈ C∞out(M, TM), then {ϕs} ∈ Isout. If X ∈ C∞tan(M, TM), then

{ϕs} ∈ Istan.

2.1.2 Varifolds and restrictions

Varifolds are fundamental in any min-max construction, compared to other general-

ized surfaces like currents, because they do not allow for cancellation of mass. In this

section, we discuss some basic facts about varifolds. For a more detailed discussion

of varifolds, one can refer to Allard [1], Simon [51], Lin-Yang02 [34] and Colding-De

Lellis [9].

A k-varifold V on M is a finite nonnegative Borel measure on the Grassmannian

of unoriented k-planes on M . Any k-varifold V gives a unique measure ‖V ‖ on M

defined by ‖V ‖(B) = V (G(B)), where G(B) is the k-Grassmannian over any Borel set

B ⊂ M . The support of ‖V ‖, denoted by supp(‖V ‖), is the smallest closed set outside

which ‖V ‖ vanishes identically. The number ‖V ‖(B) will be called the mass of V in

B. Let V(M) denote the space of 2-varifolds on M endowed with the weak topology

(see [51]), and V(M) ⊂ V(M) be the subspace of 2-varifolds supported in M . There is

a restriction map (·)xM : V(M)→ V(M) ⊂ V(M) defined by V xM(B) = V (B∩G(M))

for any B ⊂ G(M), where G(M) and G(M) denote the 2-Grassmannian over M and

M respectively. Unfortunately, the restriction map is not continuous in the weak

topology. It is only “upper semi-continuous” in the following sense: if Vi is a sequence


of varifolds in V(M) converging weakly to V , then lim supi→∞ ‖Vi‖(M) ≤ ‖V ‖(M).

This follows directly from the compactness of M and (2.6.2 (c)) in Allard [1]. The

following lemma shows that if we have equality limi→∞ ‖Vi‖(M) = ‖V ‖(M), then

VixM converges weakly to V xM as well.

Lemma 2.1.1. Let V ∈ V(M). Suppose Vi ∈ V(M) is a sequence of varifolds con-

verging weakly to V as i → ∞. If the masses ‖Vi‖(M) converges to ‖V ‖(M) as

i→∞, then the restricted varifolds VixM converges weakly to V xM as i→∞.

Proof. Since Vi is a weakly convergent sequence, ‖Vi‖(M) is bounded. Hence ‖VixM‖(M)

is also bounded. After passing to a subsequence (we use the same index i for our con-

venience), VixM converges weakly to some W ∈ V(M). Since V(M) is closed in V(M),

we have W ∈ V(M). We will show that W = V xM . This clearly proves our lemma,

since any subsequence of VixM has another subsequence converging weakly to V xM .

First, we claim that W ≤ V xM , i.e. W (f) ≤ V xM(f) for any nonnegative con-

tinuous function f on G(M). Since G(M) is a closed subset of G(M), there exists a

decreasing sequence of continuous functions φk on G(M) with 0 ≤ φk ≤ 1, φk = 1 on

G(M) and φk converges pointwise to the characteristic function χG(M) of G(M). As

Vi converges weakly to V in M , we have for each k,

limi→∞

∫G(M)

φkfdVi =

∫G(M)

φkfdV. (2.1)

Since φk and f are nonnegative, for each i and k,∫G(M)

φkfdVixM≤∫G(M)

φkfdVi (2.2)

Holding k fixed and taking i→∞ in (2.2), by (2.1), we have∫G(M)

φkfdW ≤∫G(M)

φkfdV. (2.3)

Since W is supported in M and φk = 1 on M for each k, we get

W (f) =

∫G(M)

fdW =

∫G(M)

φkfdW ≤∫G(M)

φkfdV. (2.4)


Now, since φkf converges pointwise to χG(M)f monotonically, by the monotone con-

vergence theorem,

limk→∞

∫G(M)

φkfdV =

∫G(M)

χMfdV =

∫G(M)

fdV = V xM(f). (2.5)

This proves our claim that W ≤ V xM .

Now, we want to show that W = V xM . Since we already have the inequality

W ≤ V xM . It suffices to show that ‖W‖(M) = ‖V xM‖(M). It follows from the

assumption that ‖Vi‖(M) converges to ‖V ‖(M) as i→∞,

‖W‖(M) = limi→∞‖VixM‖(M) = lim

i→∞‖Vi‖(M) = ‖V ‖(M) = ‖V xM‖(M). (2.6)

Thus, Lemma 2.1.1 is established.

2.1.3 First variation formula and its consequences

Let V ∈ V(M). Take any tangential vector fieldX ∈ C∞tan(M, TM) and let {ϕs}s∈(−ε,ε)

be the 1-parameter family of diffeomorphisms generated by X. Since ϕs(M) = M for

all s, ‖(ϕs)]V ‖(M) is a smooth function in s. Differentiating in s at s = 0, the same

calculation as that in the usual first variation formula (see [51] for example) shows

that

δMV (X) =d

ds

∣∣∣∣s=0

‖(ϕs)]V ‖(M) =

∫(x,S)∈G(M)

divS X(x) dV (x, S) (2.7)

Notice that we are only integrating over G(M) instead of G(M) on the right hand

side of (2.7) since we are only counting area in M .

Definition 2.1.2. Let U ⊂ M be an open set. A varifold V ∈ V(M) is said to

be tangentially stationary in U if and only if δMV (X) = 0 for all tangential vector

fields X ∈ C∞tan(M, TM) supported in U . We denote the set of varifolds which are

tangentially stationary in U by V∞,U ⊂ V(M). When U = M , we simply say that V

is tangentially stationary and the set of tangentially stationary varifolds is denoted

by V∞.


Remark 2.1.3. It is obvious that δMV (X) = δMV xM(X) for any V ∈ V(M) and

X ∈ C∞tan(M, TM). Therefore, if V ∈ V(M) is tangentially stationary in U , then so

is V xM and vice versa. So, we often assume that tangentially stationary varifolds are

supported in M .

Using the compactness of mass bounded varifolds in the weak topology and (2.7), it

is immediate that the set of mass bounded tangentially stationary varifolds supported

in M is compact in the weak topology.

Lemma 2.1.4. For any open set U ⊂ M , and any constant C > 0, the set

VC∞,U(M) = {V ∈ V∞,U(M) : ‖V ‖(M) ≤ C}

is compact in the weak topology.

Proof. Let Vi be a sequence of varifolds in VC∞,U(M). Since Vi are supported in

M , ‖Vi‖(M) = ‖Vi‖(M) which are uniformly bounded by C, a subsequence of

Vi (after relabeling) converges weakly to V by compactness of mass bounded vari-

folds. Since V(M) is closed, V is also supported in M , i.e. V ∈ V(M). Therefore,

‖V ‖(M) = ‖V ‖(M) = lim ‖Vi‖(M) = lim ‖Vi‖(M) ≤ C. It remains to show that

V is tangentially stationary in U , but this follows directly from the first variation

formula (2.7) and the fact that Vi and V are supported in M .

Next, we recall some useful results from Gruter-Jost [21], where the monotonicity

formula and the Allard regularity for tangentially stationary varifolds are proved.

Their proof was given only for rectifiable varifolds in RN but they can be easily

generalized to general varifolds in Riemannian manifolds.

Consider a small neighborhood U ⊂ M of ∂M where the nearest point projection

π : U → ∂M is well-defined and smooth. For each x ∈ U , we define x, the reflection

of x across ∂M , to be the unique point in U such that d(x, ∂M) = d(x, ∂M) and

π(x) = π(x). Given any subset B ⊂ U , we define its reflection across ∂M to be the

set B = {x ∈ U : x ∈ B}. Using these notations, we can now state the monotonicity

formula for stationary varifolds with free boundaries.


Theorem 2.1.5 (Gruter-Jost [21]). Let V ∈ V(M) be a tangentially stationary var-

ifold supported in M . Then for each x ∈ U , there exists r = r(x) > 0 (where

r < d(x, ∂U)), depending on x, and a constant C = C(x, r) > 0, depending on x and

r, such that for all 0 < σ < ρ < r,

‖V ‖(Bσ(x)) + ‖V ‖(Bσ(x))

πσ2≤ C(x, r)

‖V ‖(Bρ(x)) + ‖V ‖(Bρ(x))

πρ2. (2.8)

Furthermore, the constant C(x, r) can be chosen such that for each x ∈ U , limr→0C(x, r) ↓1.

Note that the above monotonicity formula reduces to the usual monotonicity for-

mula for interior points when d(x, ∂M) < σ < ρ < r.

The above theorem tells us that the density function θ : M → R

θ(x) = limρ→0

‖V ‖(Bρ(x))

πρ2(2.9)

is well defined everywhere for a tangentially stationary varifold, even at x ∈ ∂M .

Theorem 2.1.6 (Gruter-Jost [21]). Let V ∈ V(M) be a tangentially stationary var-

ifold supported in M . Let x ∈supp(‖V ‖) ∩ ∂M , and suppose θ ≥ 1 ‖V ‖-almost

everywhere. Suppose that there is a sufficiently small ρ > 0 such that

‖V ‖(Bρ(x))

πρ2≤ 1

2(1 + ε) (2.10)

for some small ε ∈ (0, 1). Then, there exists a small γ ∈ (0, 1) such that supp(‖V ‖)∩Bγρ(x) is the graph of a function u which is C1,α up to the free boundary.

In Gruter-Jost [23], the curvature estimates for stable minimal surfaces of Schoen

[49] were generalized to the free boundary case. This gives a compactness theorem

for stable minimal surfaces with free boundary.

Theorem 2.1.7 (Gruter-Jost [23]). Let U ⊂ M be an open set. Suppose {Σn} is a

sequence of stable minimal surfaces in U ∩M with free boundary on U ∩ ∂M , and

their areas are uniformly bounded. Then, for any compact subset K ⊂⊂ U , there is


a subsequence of Σn converging smoothly to a stable minimal surface Σ∞ in K ∩Mwith free boundary on U ∩ ∂M .

2.2 The min-max construction

In this section, we describe in detail the min-max construction.

2.2.1 Sweep-outs

First, we define a generalized family of surfaces which allow mild singularities and

changes in topology. We will always parametrize a sweep-out by the letter t over the

interval [0, 1] unless otherwise stated.

Definition 2.2.1. A family {Σt}t∈[0,1] of surfaces in M is said to be a generalized

smooth family of surfaces, or simply a sweep-out, if and only if there exists a finite

subset T ⊂ [0, 1] and a finite set of points P ⊂ M such that

(1) for t /∈ T , Σt is a smoothly embedded closed surface (not necessarily connected)

in M ,

(2) for t ∈ T , Σt \P is a smoothly embedded surface (not necessarily connected) in

M and Σt is compact,

(3) Σt varies smoothly in t (see Remark 2.2.2 below).

If, in addition to (1)-(3) above,

(4) H2(Σt ∩M) is a continuous function in t ∈ [0, 1], here H2 is the 2-dimensional

Hausdorff measure induced by the metric on M ,

we say that {Σt}t∈[0,1] is a continuous sweep-out.

Remark 2.2.2. The “smoothness” condition of (3) means the following: for each t /∈ T ,

for τ close enough to t, Στ is a graph over Σt (hence diffeomorphic to Σt) and Στ

converges smoothly to Σt as a graph when τ → t. At t ∈ T , for any ε > 0 small,

let Pε = {x ∈ M : d(x, P ) < ε}, then Στ \ Pε converges smoothly to Σt \ Pε in the

graphical sense above as τ → t.


Remark 2.2.3. Note that condition (4) is not redundant since we could have a con-

tinuous (or even smooth) family of Σt ⊂ M such that H2(Σt ∩M) is discontinuous.

“Generically” this would not occur, but the mass of a surface inside M can drop

suddenly when part of the surface (with positive area) sticks to the boundary ∂M

and then moves out of M in a smooth fashion. In fact, Lemma 2.1.1 implies that

condition (4) is equivalent to saying that {Σt ∩M}t∈[0,1] is a continuous family as

varifolds in M .

2.2.2 The min-max construction

Given a sweep-out {Σt}, we can deform the sweep-out to get another sweep-out by

the following procedure. Let ψ = ψt(x) = ψ(t, x) : [0, 1] × M → M be a smooth

map such that for each t ∈ [0, 1], there exists isotopies {ϕts}s∈[0,1] ∈ Isout such that

ϕt1 = ψt. We define a new family {Σ′t} by Σ′t = ψt(Σt). A collection of sweep-outs Λ

is saturated if it is closed under these deformations of sweep-outs.

Remark 2.2.4. For technical reasons, we will require that any saturated collection

Λ we consider has the additional property that there exists some natural number

N = N(Λ) <∞ such that for any {Σt} ∈ Λ, the set P in Definition 2.2.1 consists of

at most N points.

We will apply our min-max construction to a saturated collection of sweep-outs.

Given any such collection Λ, and any sweep-out {Σt} ∈ Λ, we denote by F({Σt})the area in M of its maximal slice and by m0(Λ) the infimum of F taken over all

sweep-outs of Λ; that is,

F({Σt}) = supt∈[0,1]

H2(Σt ∩M), (2.11)

and

m0(Λ) = inf{Σt}∈Λ

F({Σt}). (2.12)

Note that we have to take “sup” in the definition of F instead of “max” (as in

Colding-De Lellis [9]) because the maximum is not guaranteed to be achieved unless

the sweep-out is continuous (in the sense of Definition 2.1.1).


Definition 2.2.5. (1) A sequence {Σnt }n∈N of sweep-outs in Λ is said to be a min-

imizing sequence of sweep-outs if and only if limn→∞ F({Σnt }) = m0(Λ).

(2) Let {Σnt }n∈N be a minimizing sequence of sweep-outs, and a sequence tn ∈ [0, 1].

If limn→∞H2(Σntn ∩M) = m0(Λ), then {Σn

tn ∩M}n∈N is said to be a min-max

sequence of surfaces.

Our goal is to show that there exists some min-max sequence Σntn ∩M converging

(in the varifold sense) to a smooth compact embedded minimal surface Σ in M with

free boundary ∂Σ ⊂ ∂M (multiplicities allowed). In order to produce something

non-trivial, we require m0(Λ) > 0. We first show that this can be done by choosing

the initial sweep-out to be the level sets of a Morse function and an isoperimetric

inequality argument.

Proposition 2.2.6. There exists a saturated collection Λ of sweep-outs with m0(Λ) >

0.

Proof. Take any Morse function f : M → [0, 1] on the closed 3-manifold M . Define

Σt = f−1(t) for t ∈ [0, 1]. It is straightforward to check that {Σt}t∈[0,1] is a sweep-out

in the sense of Definition 2.1.1. Let Λ be the saturation of {Σt}, i.e. the smallest

collection of sweep-outs which is saturated and contains {Σt}. We will show that for

such a collection Λ, we have m0(Λ) > 0.

Let Ut = f−1([0, t)), and U ′t = M \ Ut. Let ψ = ψt(x) = ψ(t, x) : [0, 1]× M → M

be a smooth map such that for each t ∈ [0, 1], there exists isotopies {ϕts}s∈[0,1] ∈ Isout

such that ϕt1 = ψt. Define the new sweep-out {Γt} ∈ Λ by Γt = ψt(Σt). We claim

that F({Γt}) ≥ C > 0, where C is a constant independent of ψ. This would imply

m0(Λ) ≥ C > 0.

To prove our claim, take Vt = ψt(Ut) and V ′t = ψt(U′t). M is a disjoint union

of Vt ∩ M and V ′t ∩ M , with ∂Vt∩ int(M) = ∂V ′t∩ int(M). Since the function

t 7→ H3(Vt ∩M) is continuous, and H3(V0 ∩M) = 0, H3(V1 ∩M) = Vol(M), there

exists t0 ∈ (0, 1) such that H3(Vt0 ∩M) = 12Vol(M).

By the isoperimetric inequality, there exists a constant C = C(M) > 0 such that

1

2Vol(M) = H3(Vt0 ∩M) ≤ C(M)(H2(Γt0 ∩M))

32 . (2.13)


Hence,

F({Γt}) = supt∈[0,1]

H2(Γt ∩M) ≥(

Vol(M)

2C(M)

) 23

> 0. (2.14)

This proves our claim and thus the proposition.

2.2.3 The main result and a technical lemma

We now state the main result of this chapter.

Theorem 2.2.7. Let M be a compact Riemannian 3-manifold with boundary ∂M 6= ∅,and M ⊂ M where M is a closed Riemannian 3-manifold. Then,

(a) given any saturated collection of sweep-outs Λ, there exists a min-max sequence

of surfaces {Σntn ∩ M}n∈N obtained from Λ, which converges in the sense of

varifolds to an integer-rectifiable varifold V in M with ‖V ‖(M) = m0(Λ).

(b) Furthermore, there exists natural numbers n1, . . . , nk and smooth compact em-

bedded minimal surfaces Γ1, . . . ,Γk such that V =∑k

i=1 niΓi and each Γi either

is closed or meets ∂M orthogonally along the free boundary ∂Γi.

Section 2.3 to 2.6 is devoted to the proof of Theorem 2.2.7. As a direct corollary of

Proposition 2.2.6 and Theorem 2.2.7, we obtain the existence of embedded minimal

surfaces with (possibly empty) free boundary (first half of Main Theorem A). The

genus bound will be addressed in Section 2.7.

We end this section by proving a technical perturbation lemma, which is crucial

in the proof of Theorem 2.3.1 in the next section. To proceed, we first prove a lemma

which says that if we use a “small” isotopy to deform a surface, its area would not

increase by too much.

Lemma 2.2.8. Let V ∈ V(M) be a varifold in M . Suppose we have an outward vector

field X ∈ C∞out(M, TM), and let {ϕs}s∈[0,1] be the outward isotopy in Isout generated

by X. Then,

‖(ϕ1)]V ‖(M) ≤ ‖V ‖(M) e‖X‖C1 . (2.15)


Here, ‖X‖C1 denotes the C1-norm of the vector field X as a smooth map X : M →TM .

Proof. By the first variation formula of area in M ,

|δV (X)| ≤∣∣∣∣∫G(M)

divπX(x) dV (x, π)

∣∣∣∣ ≤ ‖V ‖(M)‖X‖C1 . (2.16)

Hence, if we let f(s) = ‖(ϕs)]V ‖(M), then (2.16) means that

|f ′(s)| ≤ f(s)‖X‖C1 . (2.17)

Integrating (2.17) on s ∈ [0, 1], we get

f(1) ≤ f(0) e‖X‖C1 (2.18)

Hence,

‖(ϕ1)]V ‖(M) ≤ ‖V ‖(M) e‖X‖C1 . (2.19)

Now, using (2.19) and that X is an outward vector field,

‖(ϕ1)]V ‖(M) ≤ ‖(ϕ1)](V xM)‖(M) ≤ ‖V xM‖(M) e‖X‖C1 = ‖V ‖(M) e‖X‖C1 ,

where the first inequality holds because ‖(ϕ1)]V ‖(M) = ‖(ϕ1)](V xM)‖(M). This

proves Lemma 2.2.8.

We are now ready to prove the perturbation lemma.

Lemma 2.2.9. Given any sweep-out {Σt}t∈[0,1] ∈ Λ, and any ε > 0, there exists a

continuous sweep-out {Σ′t}t∈[0,1] ∈ Λ such that

F({Σ′t}) ≤ F({Σt}) + ε (2.20)

Proof. By 2.6.2(d) of Allard [1] and Lemma 2.1.1 above, it suffices to construct {Σ′t}such that

H2(Σ′t ∩ ∂M) = 0 (2.21)


for all t ∈ [0, 1]. This would imply that {Σ′t} is a continuous sweep-out. One might try

to perturb Σt using an outward isotopy so that it is transversal to ∂M . However, this

is impossible, in general, to find a smooth family of outward isotopies such that all

the perturbed surfaces are transversal to ∂M . On the other hand, we could find one

so that all but finitely many Σt is transversal to the boundary ∂M after perturbation,

and for those finitely many exceptions, there are only finitely many points at which

the perturbed Σt meets the boundary ∂M non-transversally. This would certainly

imply (2.21).

For θ > 0 sufficiently small (to be chosen later), the signed distance function

d = d(·, ∂M) is a smooth function on the open tubular neighborhood Uθ(∂M) = {x ∈M : |d(x, ∂M)| < θ} of ∂M . (We take d to be nonnegative for points in M .) Let

ν be the inward pointing unit normal to ∂M with respect to M (This is globally

defined on ∂M even when M is not orientable). For θ > 0 small enough, we have a

diffeomorphism

g(x, s) = expx(sν(x)) : ∂M × (−θ, θ)→ Uθ(∂M). (2.22)

We will need the following parametric Morse theorem: If ft : N → R is a 1-

parameter family of smooth functions on a compact (possibly with boundary) man-

ifold N with t ∈ [0, 1], and f0, f1 are Morse functions, then there exists a smooth

1-parameter family Ft : N → R such that F0 = f0, F1 = f1, and F is uniformly close

to f in the Ck-topology on functions N × [0, 1] → R. Furthermore, Ft is Morse at

all but finitely many t, at a non-Morse time, the function has only one degenerate

critical point, corresponding to the birth/death transition.

The relationship between Morse functions and transversality can be seen as fol-

lows. Let Σ be a closed surface in M . Consider the function d restricted on Σ ∩Uθ(∂M), if d : Σ ∩ Uθ(∂M)→ R is a Morse function, then Σ intersects the level sets

{x ∈M : d = c} transversally except possibly at the critical points of d, which is only

a finite set. In particular, we have H2(Σ ∩ ∂M) = 0. If d is not a Morse function,

we approximate it by a Morse function dφ in Ck norm in Σ ∩ Uθ(∂M). Without loss

of generality, we can also assume that dφ ≤ d everywhere. Let φ ≤ 0 be a smooth


extension of the function dφ − d to Uθ(∂M) (by Whitney’s extension theorem), in

such a way that ‖φ‖Ck is very small. Then, if we consider the outward vector field

X(x) = χ(x)φ(x)∇d(x) (2.23)

where 0 ≤ χ ≤ 1 is a smooth cutoff function on M such that χ = 1 on Uθ/2(∂M), χ = 0

outside Uθ(∂M) and |∇χ| ≤ 4/θ. Let {ϕs}s∈[0,1] be the outward isotopy generated by

X. Then it is clear that H2(ϕ1(Σ) ∩ ∂M) = 0 according to the discussion above.

The perturbation can be carried out for each Σt in the sweep-out. Hence, we want

to choose φt ≤ 0 on Uθ(∂M) which are small in Ck norm such that, after perturbation,

Σt intersects the boundary ∂M at a set of H2- measure zero. The only complication

is that we have to choose φt which depends smoothly on t. That is why we need the

parametric Morse theorem.

By Lemma 2.2.8, for every C > 0 and ε > 0, there exists δ = δ(C, ε) > 0 sufficiently

small (for example, take δ < ln(1 + ε/2C)) such that whenever ‖X‖C1 < δ,

‖(ϕ1)]V ‖(M) ≤ ‖V ‖(M) +ε

2(2.24)

for all V ∈ V(M) with ‖V ‖(M) ≤ C.

Fix a sweep-out {Σt}t∈[0,1] ∈ Λ and ε > 0 as in the hypothesis, since {Σt} is

a continuous family of varifolds in M , there exists a constant C > 0 such that

H2(Σt ∩ M) ≤ H2(Σt) ≤ C for all t ∈ [0, 1]. For this ε and C, choose δ > 0 so

that (2.24) holds. Moreover, assume θ > 0 is always sufficiently small so that d is a

smooth function on Uθ(∂M) and (2.22) holds.

Now, we would like to apply the parametric Morse theorem to the family of smooth

functions dt : Σt∩U θ(∂M)→ R. However, there is a little technical difficulty because

Σt ∩ U θ(∂M) are not all diffeomorphic to each other. Recall that in the definition of

a sweep-out, we have two finite sets, T ⊂ [0, 1] and P ⊂ M , at which singularities

occur. First of all, we argue that we can assume P ∩ U θ(∂M) = ∅.Suppose P ∩ ∂M 6= ∅. Since P is just a finite set, there exists 0 < ρ < θ/2 such


that ∂Mρ ∩ P = ∅. Define an outward vector field X ∈ C∞out(M, TM) by

X(x) = −ρχ(x)∇d(x), (2.25)

where 0 ≤ χ ≤ 1 is a smooth cutoff function on M such that χ = 1 on Uρ(∂M),

χ = 0 outside Uθ(∂M) and |∇χ| ≤ 4/θ. Let K > 0 be a constant (independent of

θ) so that |∇2d| ≤ K on U θ(∂M). Hence, for ρ > 0 sufficiently small (depending on

θ and K), we can make ‖X‖C1 < δ. Let {ϕs}s∈[0,1] be the isotopy generated by X,

then by (2.24),

H2(ϕ1(Σt) ∩M) ≤ H2(Σt ∩M) +ε

2. (2.26)

Moreover, ϕ1(∂Mρ) = ∂M . Therefore, replacing {Σt} by {ϕ1(Σt)} if necessary, we

can assume that P ∩ ∂M = ∅.As P is a finite set, we can further assume that θ is small enough so that P ∩

U θ(∂M) = ∅. By the definition of a sweep-out, there exists a partition 0 = t0 < t1 <

· · · < tk = 1 of the interval [0, 1] such that on each subinterval [ti−1, ti], i = 1, . . . , k,

there exists an open neighborhood Ui of ∂M contained in Uθ(∂M) such that Σt ∩ Uiare all diffeomorphic for t ∈ [ti−1, ti].

Now, for any δ′ > 0 (to be specified later), since Morse functions are dense in Ck-

topology, for each i = 0, 1, . . . , k, we can approximate the smooth function dti : Σti ∩Ui → R by a Morse function dti : Σti ∩ Ui → R, with dti ≤ dti and ‖dti − dti‖C1 < δ′.

Let φti ≤ 0 be a smooth extension of dti − dti to Ui such that ‖φti‖C1 ≤ δ′ for all i.

Using the parametric Morse Theorem, we can construct a smooth family of smooth

functions φt ≤ 0 on Ui, t ∈ [ti−1, ti] such that dt + φt is a Morse function on Σt ∩ Uiexcept for finitely many t’s there is only one degenerate critical point. We can also

assume that φt is uniformly small in C1-norm on Uθ(∂M). Putting these intervals

together, we have a piecewise smooth 1-parameter family of smooth functions φt,

defined on U for some neighborhood U of ∂M , such that dt + φt are Morse except

at finitely many times. Since Morse functions form an open set in the space of all

smooth functions in the C∞-topology, and the family is Morse at each ti, we can

smooth out the family, keeping it Morse except at finitely many times away from

ti’s. Assume θ is chosen small enough such that Uθ(∂M) ⊂ U . In summary, we have


a smooth 1-parameter family of smooth non-positive functions {φt} on U such that

‖φt‖C1 < δ′. For each t ∈ [0, 1], let {ϕt(s)}s∈[0,1] be the outward isotopy generated by

the outward vector field Xt ∈ C∞out(M, TM) defined as

Xt(x) = φt(x)χ(x)∇d(x), (2.27)

where 0 ≤ χ ≤ 1 is a smooth cutoff function on M so that χ = 1 on Uθ/2(∂M), χ = 0

outside Uθ(∂M) and |∇χ| ≤ 2/θ. Let {ϕt(s)}s∈[0,1] be the isotopy in Isout generated

by Xt. Take Σ′t = ϕt(1)(Σt), then {Σ′t} ∈ Λ. We claim that {Σ′t} is the competitor

we want.

First of all, by choosing δ′ > 0 sufficiently small, we can make ‖Xt‖C1 < δ for all

t ∈ [0, 1]. Hence, we have from (2.24) that

H2(Σ′t ∩M) ≤ H2(Σt ∩M) +ε

2. (2.28)

Moreover, since d : Σ′t ∩ Uθ(∂M) → R agrees with dt + φt for all t ∈ [0, 1], by our

construction, we have Σ′t∩∂M consists of at most finitely many points for all t ∈ [0, 1].

Therefore, we have

H2(Σ′t ∩ ∂M) = 0.

This completes the proof of Lemma 2.2.9.

2.3 Existence of stationary varifolds

As pointed out in Colding-De Lellis [9], a technical step in the proof is to “tighten”

the sweep-outs so that almost maximal slices are almost stationary. While it is

easy to get a subsequence of a min-max sequence converging to a limit varifold, using

standard compactness theorem for uniformly mass-bounded Radon measures, we need

to make sure that the limit varifold obtained is stationary. To achieve this, we have

to “tighten” our sweep-outs first.

In this section, we prove that these exists some “nice” minimizing sequence of


sweep-outs, such that any min-max sequence of surfaces obtained from such a mini-

mizing sequence has a subsequence converging to a varifold in M which is tangentially

stationary. We can even construct a minimizing sequence of continuous sweep-outs,

using the perturbation lemma (Lemma 2.2.9). This is essential in the proof of exis-

tence of almost minimizing sequence in the next section.

Theorem 2.3.1. There exists a minimizing sequence of sweep-outs {Σnt }n∈N in the

saturated collection Λ such that

(1) {Σnt }t∈[0,1] is a continuous sweep-out for each n ∈ N.

(2) Every min-max sequence {Σntn ∩M}n∈N constructed from such a minimizing se-

quence has a subsequence converging weakly to a tangentially stationary varifold

V ∈ V∞ supported in M .

Proof. Let {Σnt }n∈N ⊂ Λ be a minimizing sequence of sweep-outs. By Lemma 2.2.9,

it is clear that we can assume {Σnt }t∈[0,1] is a continuous sweep-out for each n ∈ N.

So (1) is established.

Fix some C > 4m0. By Lemma 2.1.4, VC∞(M) ⊂ VC(M) is a compact set in the

weak topology. We want to construct a “tightening” map

Ψ : VC(M)→ Istan

such that

(a) Ψ is continuous with respect to the weak topology on VC(M) and the L∞-norm

on Istan.

(b) If V ∈ VC∞(M), then Ψ(V ) is the identity isotopy on M .

(c) If V /∈ VC∞(M), then ‖(Ψ(V )1)]V ‖(M) ≤ ‖V ‖(M) − L(d(V,VC∞(M)) for some

continuous strictly increasing function L : R → R with L(0) = 0. (Here, d is

any metric on VC(M) whose metric topology agrees with the weak topology.)


In order to construct the tightening map Ψ, we are going to define tangential

isotopies Ψ(V ) which are generated by 1-parameter families of diffeomorphisms sat-

isfying certain ordinary differential equations. First of all, we associate to any V a

suitable tangential vector field, which will be used to construct Ψ(V ).

For each integer k, we define the annular neighborhood of tangentially stationary

varifolds V∞

Vk = {V ∈ VC(M) :1

2k+1≤ d(V,VC∞(M)) ≤ 1

2k}.

There exists a positive constant c(k) depending on k such that to every V ∈ Vk we

can associate a smooth tangential vector field χV ∈ C∞tan(M, TM) with ‖χV ‖∞ ≤ 1/k

for k > 0 and ‖χV ‖∞ ≤ 1 for k ≤ 0, moreover,

δMV (χV ) ≤ −c(k).

Our next task is choosing χV with continuous dependence on V . Note that for every

V there is some radius r such that δMW (χV ) ≤ −c(k)/2 for every W ∈ Ur(V ) where

Ur(V ) denotes the open ball of radius r centered at V in VC(M) with the metric d.

Hence, by compactness of Vk, for any k we can find balls {Uki }i=1,...,N(k) and tangential

vector fields χki such that

The balls Uki concentric to Uk

i with half the radii cover Vk; (2.29)

If W ∈ Uki , then δMW (χki ) ≤ −c(k)/2; (2.30)

The balls Uki are disjoint from Vj if |j − k| ≥ 2. (2.31)

Hence, {Uki }k,i is a locally finite covering of VC(M) \ VC∞(M). To this family we can

subordinate a continuous partition of unity ϕki . Thus we set HV =∑

i,k ϕki (V )χki .

Note that HV is a tangential vector field. The map H : VC(M) → C∞tan(M, TM)

which to every V associates HV is continuous. Moreover, ‖HV ‖∞ ≤ 1 for every V .

For V ∈ Vk we let r(V ) be the radius of the smallest ball U ji which contains it.

We find that r(V ) > r(k) > 0, where r(k) only depends on k. Moreover, by (2.30)


and (2.31), for every W contained in the ball Ur(V )(V ) we have that

δMW (HV ) ≤ −1

2min{c(k − 1), c(k), c(k + 1)}.

Summarizing there are two continuous functions g : R+ → R+ and r : R+ → R+ such

that

δMW (HV ) ≤ −g(d(V,VC∞(M))) if d(W,V ) ≤ r(d(V,VC∞(M))). (2.32)

Now for every V construct the continuous 1-parameter family of diffeomorphisms

ΦV : [0,+∞)× M → M with∂ΦV (t, x)

∂t= HV (ΦV (t, x)).

For each t and V , we denote by ΦV (t, ·) the corresponding diffeomorphism of M . We

claim that there are continuous functions T : R+ → [0, 1] and G : R+ → R+ such that

lims→0+ G(s) = lims→0+ T (s) = 0 and if γ = d(V,VC∞(M)) > 0 and we transform V

into V ′ via the diffeomorphism ΦV (T (γ), ·), then ‖V ′‖(M) ≤ ‖V ‖(M)−G(γ). Indeed

fix V . For every r > 0 there is a T > 0 such that the curve of varifolds

{V (t) = (ΦV (t, ·))]V : t ∈ [0, T ]}

stays in Ur(V ). Thus

‖V (T )‖(M)− ‖V ‖(M) ≤∫ T

0

[δMV (t)](HV )dt,

and therefore if we choose r = r(d(V,VC∞(M))) as in (2.32), then we get the bound

‖V (T )‖(M)− ‖V ‖(M) ≤ −Tg(d(V,VC∞(M))).

Using a procedure similar to that above we can choose T depending continuously on

V . It is then trivial to see that we can in fact choose T so that at the same time it

is continuous and depends only on d(V,VC∞(M)).


For each V , set γ = d(V,VC∞(M)) and

ΨV (t, ·) = ΦV ([T (γ)]t, ·) for t ∈ [0, 1].

Hence, ΨV is a “normalization” of ΦV . From above, we know that there is a continuous

strictly increasing function L : R+ → R+ such that L(0) = 0 and

ΨV (1, ·) transforms V into a varifold V ′ with ‖V ′‖(M) ≤ ‖V ‖(M)− L(γ),

which is property (c) of the tightening map. From the construction property (a) and

(b) follows immediately.

Therefore, such a tightening map exists. Since {Σnt } are continuous sweep-outs,

for each n, {Σnt ∩M}t∈[0,1] is a continuous family in VC(M). Therefore, Ψ(Σt ∩M) is

a continuous family in Istan. By a smoothing argument (for example, a convolution

in the t-variable), we can make it a smooth family. We use these tangential isotopies

to deform the minimizing sequence {Σnt } to another minimizing sequence {Γnt } which

satisfies

H2(Γnt ∩M) ≤ H2(Σnt ∩M)− L(d(Σn

t ,VC∞(M)))/2. (2.33)

As {Σnt }n is a minimizing sequence of sweep-outs, we can assume without loss of

generality that F({Σnt }n) ≤ m0 + 1/n. Furthermore, the sequence {Γnt } still satisfies

(1) since only tangential vector fields are used in the deformations.

Next, we claim that for every ε > 0, there exist δ > 0 and N ∈ N such that

whenever n > N and H2(Γntn ∩M) > m0− δ, we have d(Γntn ,V∞) < ε. To see this, we

argue by contradiction. Note first that the construction of the tightening map yields

a continuous and increasing function λ : R+ → R+ (independent of t and n) such

that λ(0) = 0 and

d(Σt,VC∞(M)) ≥ λ(d(Γnt ,V

C∞(M))). (2.34)

Fix ε > 0 and choose δ > 0, N ∈ N such that L(λ(ε))/2 − δ > 1/N . We claim

that for this choice of δ and N , whenever n > N and H2(Γntn ∩ M) > m0 − δ,

we have d(Γntn ,V∞) < ε. Suppose not. Then there are n > N and tn such that


H2(Γntn ∩M) > m0 − δ and d(Γnt ,VC∞(M)) > ε. Hence, from (2.33) and (2.34) we get

H2(Σnt ∩M) > H2(Γnt ∩M) +

L(λ(ε))

2> m0 +

L(λ(ε))

2− δ > m0 +

1

N> m0 +

1

n.

This contradicts the assumption that F({Σnt }n) ≤ m0 + 1/n. This proves our claim.

It is clear that the claim implies (2) and the proof is completed.

2.4 Existence of almost minimizing sequence

Even though we know from section 2.3 that any min-max sequence has a subsequence

converging to a tangentially stationary varifold , a tangentially stationary varifold

can be far from a smoothly embedded minimal surface. In order to get regularity,

we require our min-max sequence to satisfy a stronger condition, called the “almost

minimizing property”. Roughly speaking, a surface is almost minimizing means that

if you want to decrease its area in M through an isotopy, its area in M must become

large at some time during the deformation. The precise definition is given below.

Definition 2.4.1 (Almost Minimizing Property). Given ε > 0 and an open set U ⊂M , a varifold V ∈ V(M) is ε- almost minimizing in U if and only if there DOES NOT

exist isotopy {ϕs}s∈[0,1] ∈ Isout(U) such that

1. ‖(ϕs)]V ‖(M) ≤ ‖V ‖(M) + ε8

for all s ∈ [0, 1];

2. ‖(ϕ1)]V ‖(M) ≤ ‖V ‖(M)− ε.

A sequence {V n} ⊂ V(M) is said to be an almost minimizing sequence in U if each

V n is εn-almost minimizing in U for some sequence εn ↓ 0.

Remark 2.4.2. The definition is almost the same as the one used in Colding-De Lellis

[9] except that we are consider the area in M and outward isotopies only.

In this section, we prove the existence of a min-max sequence which is almost

minimizing on small annuli. We will follow the ideas of the proof given in Colding-De

Lellis [9] and make the modifications as needed. First, we recall some definitions.


Definition 2.4.3. Let CO be the set of pairs (U1, U2) of open sets in M with

d(U1, U2) > 2 min{diam(U1), diam(U2)}. (2.35)

Given (U1, U2) ∈ CO, we say that V ∈ V(M) is ε-almost minimizing in (U1, U2) if it

is ε-almost minimizing in at least one of the U1 or U2.

Remark 2.4.4. The significance of CO is that for any (U1, U2) and (V 1, V 2) ∈ CO,

there are some i, j = 1, 2 with d(U i, V j) > 0, hence U i ∩ V j = ∅.

The main theorem of this section is the following.

Theorem 2.4.5. Let {Σnt } be a minimizing sequence as given in Theorem 2.3.1.

Then, there is a min-max sequence {ΣL}L∈N = {Σn(L)tn(L)}L∈N such that

each ΣL is1

L-almost minimizing in every (U1, U2) ∈ CO. (2.36)

Proof. We argue by contradiction. First of all, we fix a minimizing sequence {Σnt }n∈N ⊂

Λ satisfying Theorem 2.3.1 and such that

F({Σnt }) < m0 +

1

n. (2.37)

Fix L ∈ N. To prove the proposition, we claim that there exists n > L and

tn ∈ [0, 1] such that Σn = Σntn satisfies

(a) Σn is 1L

-almost minimizing in every (U1, U2) ∈ CO.

(b) H2(Σn ∩M) ≥ m0 − 1L

.

We define the sets of “big slices” for each n > L by

Kn =

{t ∈ [0, 1] : H2(Σn

t ∩M) ≥ m0 −1

L

}. (2.38)

Note that Kn is compact by condition (4) in Definition 2.2.1. If the claim above is

false, then for every t ∈ Kn, there exists a pair of open subsets (U1,t, U2,t) ∈ CO such


that Σnt is not 1

L-almost minimizing in any of them. So for every t ∈ Kn, there exists

isotopies {ϕ1,ts }s∈[0,1] ∈ Isout(U1,t) and {ϕ2,t

s }s∈[0,1] ∈ Isout(U2,t) such that for i = 1, 2,

(1) H2(ϕi,ts (Σnt ) ∩M) ≤ H2(Σn

t ∩M) + 18L

for every s ∈ [0, 1];

(2) H2(ϕi,t1 (Σnt ) ∩M) ≤ H2(Σn

t ∩M)− 1L

.

Next, we want to establish the following claim. Claim: For each t ∈ Kn, there

exists δ = δ(t) > 0 such that if |τ − t| < δ, then for i = 1, 2,

(1’) H2(ϕi,ts (Σnτ ) ∩M) ≤ H2(Σn

τ ∩M) + 14L

for every s ∈ [0, 1];

(2’) H2(ϕi,t1 (Σnτ ) ∩M) ≤ H2(Σn

τ ∩M)− 12L

.

To see why (1’) is true, we argue by contradiction. Suppose no such δ exists, then

there exists a sequence τj → t and sj ∈ [0, 1] such that for all j,

H2(ϕi,tsj (Σnτj

) ∩M) > H2(Σnτj∩M) +

1

4L. (2.39)

After passing to a subsequence, we can assume that sj → s0 for some s0 ∈ [0, 1].

Observe that ϕi,tsj (Σnτj

) converges weakly as varifolds to ϕi,ts0(Σnt ) as j → ∞. By

(2.6.2(c)) in Allard [1] and the fact that the sweep-outs {Σnt } are continuous, we

have

H2(ϕi,t(Σnt ) ∩M) ≥ lim sup

j→∞H2(ϕi,tsj (Σ

nτj

) ∩M)

≥ limj→∞

H2(Σnτj∩M) +

1

4L

= H2(Σnt ∩M) +

1

4L.

This contradicts (1) above. So we can choose δ > 0 such that (1’) holds.

The proof of (2’) is similar. Again, if no such δ > 0 exists, then there exists a

sequence τj → t such that for all j,

H2(ϕi,t1 (Σnτj

) ∩M) > H2(Σnτj∩M)− 1

2L. (2.40)


Since ϕi,t1 (Σnτj

) converges weakly to ϕi,t1 (Σnt ) in M as j →∞, thus, we have

H2(ϕi,t1 (Σnt ) ∩M) ≥ lim sup

j→∞H2(ϕi,t1 (Σn

τj) ∩M)

≥ limj→∞

H2(Σnτj∩M)− 1

2L

= H2(Σnt ∩M)− 1

2L.

This contradicts (2) above. Therefore, (2’) holds for some δ > 0.

By compactness of Kn we can cover Kn with a finite number of intervals satisfying

(1’) and (2’). This covering {Ik} can be chosen so that Ik overlaps only with Ik−1 and

Ik+1. Summarizing we can find

(i’) closed intervals I1, . . . , Ir whose interiors cover Kn,

(ii’) pairs of open sets (U1,1, U2,1), . . . , (U1,r, U2,r) ∈ CO; and

(iii’) pairs of isotopies (ϕ1,1, ϕ2,1), . . . , (ϕ1,r, ϕ2,r)

such that for i = 1, 2, and j = 1, . . . , r,

(A’) Ij ∩ Ik = ∅ if |k − j| ≥ 2;

(B’) ϕi,j is supported in U i,j;

(C’) H2(ϕi,js (Σnt ) ∩M) ≤ H2(Σn

t ∩M) + 14L

for every s ∈ [0, 1] and every t ∈ Ij,

(D’) H2(ϕi,j1 (Σnt ) ∩M) ≤ H2(Σn

t ∩M)− 12L

for every t ∈ Ij.

Now, we want to find a covering {J1, . . . , JR} which is a refinement of {I1, . . . , Ir}such that we have

(i) closed intervals J1, . . . , JR in [0, 1] whose interior covers Kn,

(ii) open sets V1, . . . , VR among {U i,j}, and

(iii) isotopies ψ1, . . . , ψR among {ϕi,j}

such that


(A) Ji ∩ Jk = ∅ for |k − j| ≥ 2; if Ji ∩ Jk 6= ∅, then d(Vi, Vk) > 0;

(B) ψi is supported in Vi;

(C) H2(ψis(Σnt ) ∩M) ≤ H2(Σn

t ∩M) + 14L

for every s ∈ [0, 1] and every t ∈ Ji;

(D) H2(ψi1(Σnt ) ∩M) ≤ H2(Σn

t ∩M)− 12L

for every t ∈ Ji.

We start by setting J1 = I1, and we distinguish two cases

Case a1 : I1 ∩ I2 = ∅; then we set V1 = U1,1 and ψ1 = ϕ1,1.

Case a2 : I1∩I2 6= ∅; by Remark 2.4.4 we can choose i, k ∈ {1, 2} such that d(U1,i, U2,k) >

0 and we set V1 = U1,i and ψ1 = ϕ1,i.

We now come to the choice of J2 and J3. If we come from Case a1, then:

Case b1 : We make our choice as above replacing I1 and I2 with I2 and I3;

If we come from Case a2. then we let i and k be as above and we further distinguish

two cases.

Case b21 : I2 ∩ I3 = ∅; we define J2 = I2, V2 = U2,k and ψ2 = ϕ2,k.

Case b22 : I2∩I3 6= ∅; by Remark 2.4.4 there exists l,m ∈ {1, 2} such that d(U l,2, Um,3) >

0. If l = k, then we define J2 = I2, V2 = Uk,2 and ψ2 = ϕk,2. Otherwise we

choose two closed intervals J2, J3 ⊂ I2 such that their interior cover the interior

of I2; J2 does not overlap with any Ih for h 6= 1, 2; and J3 does not overlap with

any Ih for h 6= 2, 3. Thus we set V2 = Uk,2, ψ2 = ϕk,2, and V3 = U l,2, ψ3 = ϕl,2.

An inductive argument using this procedure gives the desired covering. Note that

that cardinality of {J1, . . . , JR} is at most 2r − 1.

Choose smooth cutoff functions ηi : R→ [0, 1] supported in Ji such that for every

t ∈ Kn, there exists ηi with ηi(t) = 1. Fix t ∈ [0, 1] and let Indt be the set of all i

such that t ∈ Ji. Define for each t ∈ [0, 1],

Γnt =

{ψiηi(t)(Σ

nt ) in the open sets Vi, i ∈ Indt,

Σnt otherwise

(2.41)


In view of (A) and (B), {Γnt } is well defined and belongs to Λ. Note that by (A)

every Indt consists of at most two integers. Assume for the sake of argument that

Indt consists of exactly two integers. From the construction, there exists si, sk ∈ [0, 1]

such that Γnt is obtained from Σnt via the diffeomorphisms ψi(si, ·), ψk(sk, ·). By (A)

these diffeomorphisms are supported on disjoint sets. Thus if t /∈ Kn, then (D) gives

H2(Γnt ∩M) ≤ H2(Σnt ∩M) +

2

4L≤ m0 −

1

2L.

If t ∈ Kn, then at least one of si, sk is equal to 1. Hence (C) and (D) give

H2(Γnt ∩M) ≤ H2(Σnt ∩M)− 1

2L+

1

4L≤ F({Σn

t })−1

4L.

Therefore, F({Γnt }) ≤ F({Σnt })− 1

4L, which implies lim infn F({Γnt }) < m0, a contra-

diction.

For any x ∈ M , 0 < s < t, let An(x, s, t) denote the open annulus centered at x

with inner radius s and outer radius t. Let r > 0, we set ANx(r) as the collection

of open annuli An(x, s, t) such that 0 < s < t < r. By the same argument as in

Proposition 5.1 of Colding-De Lellis [9], we obtain the following result.

Theorem 2.4.6. There exists a positive function r : M → R and a min-max sequence

{Σj} such that

(1) {Σj} is an almost minimizing sequence in every An ∈ ANr(x)(x), for all x ∈ M ;

(2) In every such An, Σj is a smooth surface when j is sufficiently large;

(3) Σj ∩M converges to a tangentially stationary varifold V in M as j ↑ ∞.

Proof. We claim that a subsequence of the ΣL’s of Theorem 2.4.5 satisfies the re-

quirements of Theorem 2.4.6. Indeed fix L ∈ N and r > 0 such that the injectivity

radius of M is greater than 4r. Since (Br(x), M \ B4r(x)) ∈ CO we then know that

ΣL is 1/L-almost minimizing in (Br(x), M \B4r(x)). Thus we have that

either ΣL is 1/L-almost minimizing on Br(y) for every y ∈ M (2.42)

or there is xLr ∈ M such that ΣL is 1/L-almost minimizing on M \B4r(xLr ). (2.43)


If for some r > 0 there exists a subsequence {Σj} satisfying (2.42), then we are

done. Otherwise we may assume that there are two sequences of natural numbers

n ↑ ∞ and j ↑ ∞, and points xnj such that xnj → xj for n ↑ ∞ and xj → x

for j ↑ ∞. Furthermore, for every j, and for n large enough, Σn is 1/n-almost

minimizing in M \B1/j(xnj ). Of course, if U ⊂ V and Σ is ε-almost minimizing in V ,

then Σ is ε-almost minimizing in U . Since B1/j(xnj ) ⊂ B2/j(x) for n, j large, we have

M \ B2/j(x) ⊂ M \ B1/j(xnj ). Thus for every j large, the sequence {Σn} is almost

minimizing in M \B2/j(x). Together with Theorem 2.3.1, this proves that there exists

a subsequence {Σj} which satisfies conditions (1) and (3) in Theorem 2.4.6 for some

positive function r : M → R+.

It remains to show that an appropriate further subsequence satisfies (2). Each Σj

is smooth except at finitely many points. We denote by Pj the set of singular points

of Σj. After extracting another subsequence we can assume that Pj is converging, in

the Hausdorff topology, to a finite set P (here we have to use Remark 2.2.4). If x ∈ Pand An is any annulus centered at x, then Pj ∩ An = ∅ for j large enough. If x /∈ Pand An is any small annulus centered at x with outer radius less than d(x, P ), then

Pj ∩An = ∅ for j large enough. Thus, after possibly modifying the function r above,

the sequence {Σj} satisfies (1)-(3) above.

2.5 A minimization problem with partially free bound-

ary

In this section, we prove a result about minimizing area in M among isotopic surfaces

similar to the ones obtained by Almgren-Simon [2], Meeks-Simon-Yau [35], Gruter-

Jost [23] and Jost [30]. Since we are restricting to the class of outward isotopies, we

need to modify some of the arguments used in the papers mentioned above.

First, we define the concept of admissible open sets.

Definition 2.5.1. An open set U ⊂ M is said to be admissible if it satisfies all the

following properties:

(i) U is smooth, i.e. U is an open set with smooth boundary ∂U ;


(ii) U is uniformly convex in the sense that all the principal curvatures with respect

to the inward normal is positive along ∂U ;

(iii) the closure U is diffeomorphic to the closed unit 3-ball in R3;

(iv) ∂U intersects ∂M transversally and U ∩ ∂M is topologically an open disk;

(v) the angle between ∂U and ∂M is always less than π/2 when measured in U∩M ,

i.e. if νU and νM are the outward unit normal of U and M respectively, then

νU · νM < 0 along ∂U ∩ ∂M .

Given a surface Σ in M , we want to minimize area (in M) among all the surfaces

which are outward isotopic to Σ and are identical to Σ outside an admissible open

set U .

Definition 2.5.2. Let U ⊂ M be an admissible open set. Let Σ ⊂ M be an embedded

closed surface (not necessarily connected) intersecting ∂U transversally. Consider the

minimization problem (Σ, Isout(U)) and let

α = inf{ϕs}∈Isout(U)

H2(ϕ1(Σ) ∩M). (2.44)

If a sequence {ϕks}k∈N ∈ Isout(U) satisfies

limk→∞

H2(ϕk1(Σ) ∩M) = inf{ϕs}∈Isout(U)

H2(ϕ1(Σ) ∩M) = α, (2.45)

we say that Σk = ϕk1(Σ) is a minimizing sequence for the minimization problem

(Σ, Isout(U)).

Note that if two surfaces Σ1 and Σ2 agree in M , i.e. Σ1 ∩M = Σ2 ∩M , then

the minimization problems (Σ1, Isout(U)) and (Σ2, Is

out(U)) are completely identical

since we only count area in M and points outside M cannot get back to M using

outward isotopies.

From the proof of the perturbation lemma (Lemma 2.2.9), it is easy to see that

we can always perturb a minimizing sequence Σk to another minimizing sequence Σ′k


so that Σ′k intersects ∂M transversally for every k. The key result of this section is

the following theorem.

Theorem 2.5.3. Let U ⊂ M be an admissible open set, suppose {Σk} is a minimizing

sequence for the minimization problem (Σ, Isout(U)) defined in Definition 2.5.2 so that

(i) Σk intersects ∂M transversally for each k, and

(ii) Σk ∩M converge weakly to a varifold V ∈ V(M).

Then,

(a) V = Γ, for some compact embedded minimal surface Γ ⊂ U ∩M with smooth

boundary (except possibly at ∂U ∩ ∂M) contained in ∂(U ∩M);

(b) The fixed boundary of Γ is the same as Σ ∩ (∂U ∩M);

(c) Γ meets ∂M ∩ U orthogonally along the free boundary of ∂Γ;

(d) Γ is stable with respect to all tangential variations supported in U .

Remark 2.5.4. It is clear that V is a tangentially stationary varifold in M . The

only thing we have to prove is regularity. Interior regularity follows from the local-

ized version of Meeks-Simon-Yau [35] given in Proposition 3.3 of De Lellis-Pellandini

[13]. The regularity at the fixed boundary ∂U∩int(M) is also discussed in De Lellis-

Pellandini [13]. Therefore, the only case left is the regularity at the free boundary

∂M ∩ U . Hence, Theorem 2.5.3 says that the limit varifold V is equal to a stable

smoothly embedded minimal surface Γ (possibly disconnected).

The proof of Theorem 2.5.3 goes as follows. We first apply a version of local

γ-reduction (see Meeks-Simon-Yau [35] and De Lellis-Pellandini [13]) to reduce the

minimization problem to the case of genus zero surfaces. Then we use a result in Jost

[30] to show regularity of such minimizers.


2.5.1 Local γ-reductions

Following De Lellis-Pellandini [13] and Meeks-Simon-Yau [35], replacing area by area

in M , we collect these modified definitions and propositions for our purpose. First of

all, we fix δ > 0 such that the following lemma holds (see Lemma 4.2 of Jost [30]).

Lemma 2.5.5. There exists r0 > 0 and δ ∈ (0, 1), depending only on M and M , with

the property that if Σ is a surface in int(M) with ∂Σ ⊂ ∂M and

H2(Σ ∩Br0(x)) < δ2r20 for each x ∈ M (2.46)

then there exists a unique compact set K ⊂M such that

(a) ∂K∩ int(M) = Σ (i.e. K is bounded by Σ modulo ∂M);

(b) H3(K ∩Br0(x)) ≤ δ2r30 for each x ∈ M ; and

(c) H3(K) ≤ c0H2(Σ)

32 , where c0 depends only on M and M .

By rescaling the metric of M if necessary, we can assume that r0 = 1 in Lemma

2.5.5. From now on, we will assume that δ > 0 satisfies Lemma 2.5.5 with r0 = 1.

Suppose 0 < γ < δ2/9.

Definition 2.5.6. Let Σ1 and Σ2 be closed (maybe disconnected) embedded surfaces

in M . We say that Σ2 is a (γ, U)-reduction of Σ1 and write

Σ2

(γ,U)� Σ1

if the following conditions are satisfied:

(1) Σ2 is obtained from Σ1 through a surgery in U , that is,

(i) Σ1 \ Σ2 ∩M = A ⊂ U is diffeomorphic to either a closed annulus A =

{(x1, x2) ∈ R2|1 ≤ x21 + x2

2 ≤ 2} or a closed half-annulus A+ = {(x1, x2) ∈A|x2 ≥ 0};


(ii) Σ2 \ Σ1 ∩ M = D1 ∪ D2 ⊂ U with each Di diffeomorphic to either the

closed unit disk D = {(x1, x2) ∈ R2|x21 + x2

2 ≤ 1} or the closed unit half-

disk D+ = {(x1, x2) ∈ D|x2 ≥ 0};

(iii) There exists a compact set Y embedded in U , homeomorphic to the closed

unit 3-ball with ∂Y = A∪D1∪D2 modulo ∂M (i.e. there exists a compact

set K ⊂ ∂M such that ∂Y = A∪D1∪D2∪K), and (Y \∂Y )∩(Σ1∪Σ2) = ∅.

(2) H2(A) + H2(D1) + H2(D2) < 2γ;

(3) If Γ is a connected component of Σ1 ∩ U ∩ M containing A, and Γ \ A is

disconnected, then for each component of Γ \ A we have one of the following

possibilities:

(a) either it is a genus 0 surface contained in U ∩M with area ≥ δ2/2;

(b) or it is not a genus 0 surface.

We say that Σ is (γ, U)-irreducible if there does not exist Σ such that Σ(γ,U)� Σ.

A immediate consequence of the above definition is the following.

Proposition 2.5.7. Σ is (γ, U)-irreducible if and only if whenever ∆ ⊂ U ∩M is a

closed disc or half-disk with ∂∆ \∂M = ∆∩Σ and H2(∆) < γ, then there is a closed

genus 0 surface D ⊂ Σ ∩ U ∩M with ∂∆ \ ∂M = ∂D \ ∂M and H2(D) < δ2/2.

Similarly, we define a strong (γ, U)-reduction as follows.

Definition 2.5.8. Let Σ1 and Σ2 be closed (maybe disconnected) embedded surfaces

in M . We say that Σ2 is a strong (γ, U)-reduction of Σ1 and write

Σ2

(γ,U)< Σ1

if there exists an isotopy {ψs}s∈[0,1] ∈ Isout(U) such that

(1) Σ2

(γ,U)� ψ1(Σ1);

(2) Σ2 ∩ (M \ U) = Σ1 ∩ (M \ U);


(3) H2((ψ1(Σ1)∆Σ1) ∩M) < γ.

We say that Σ is strongly (γ, U)-irreducible if there is no Σ such that Σ(γ,U)< Σ.

Following the same arguments in Remark 3.1 of Meeks-Simon-Yau [35], we have

the following proposition.

Proposition 2.5.9. Given any closed embedded surface Σ (not necessarily con-

nected), there exists a sequence Σ = Σ1,Σ2, . . . ,Σk of closed embedded surfaces

(not necessarily connected) such that

Σk

(γ,U)< Σk−1

(γ,U)< · · ·

(γ,U)< Σ1 = Σ (2.47)

and

Σk is strongly (γ, U)-irreducible. (2.48)

Furthermore, there exists a constant c > 0 which depends only on genus(Σ∩M) and

H2(Σ ∩M)/δ2 so that

k ≤ c, (2.49)

and

H2((Σ∆Σk) ∩M) ≤ 3cγ. (2.50)

The following theorem gives our main result for strongly (γ, U)-irreducible surfaces

Σ. For any closed surface Σ, we denote

E(Σ) = H2(Σ ∩M)− infΣ′∈JU (Σ)

H2(Σ′ ∩M), (2.51)

where JU(Σ) = {ϕ1(Σ) : {ϕs}s∈[0,1] ∈ Isout(U)}. Let Σ0 denote the union of all

components Λ ⊂ Σ ∩ U ∩M such that there exists some KΛ ⊂ U diffeomorphic to

the unit 3-ball such that Λ ⊂ KΛ and ∂KΛ ∩ Σ ∩M = ∅.

Theorem 2.5.10. Let U ⊂ M be an admissible open set, and A ⊂ U be a com-

pact subset diffeomorphic to the unit 3-ball. Assume ∂M intersects both ∂U and ∂A

transversally.


Suppose Σ ⊂ M is a smooth closed embedded surface (possibly disconnected) such

that

(i) Σ intersects both ∂M and ∂A transversally;

(ii) E(Σ) ≤ γ/4 and is strongly (γ, U)-irreducible;

(iii) For each component Γ of Σ∩∂A∩M , let FΓ be a the component in (∂A∩M)\Γ

such that ∂FΓ \ ∂M = Γ and H2(FΓ) = min{H2(FΓ),H2((∂A ∩ M) \ FΓ)}.Furthermore, suppose that

∑qj=1 H

2(Fj) ≤ γ8, where Fj = FΓj and Γ1, . . . ,Γq

denote the components of Σ∩∂A∩M . Note that each Γ is either a closed Jordan

curve in M or a Jordan arc with endpoints on ∂M , and each FΓ is either a disk,

a half-disk or an annulus in M .

Then, H2(Σ0) ≤ E(Σ) and there exists pairwise disjoint, connected, closed genus 0

surfaces D1, . . . , Dp with Di ⊂ (Σ\Σ0)∩U∩M , ∂Di\∂M ⊂ ∂A and (∪pi=1Di)∩A∩M =

(Σ \ Σ0) ∩ A ∩M . Moreover,

p∑i=1

H2(Di) ≤q∑j=1

H2(Fj) + E(Σ), (2.52)

Furthermore, for any given α > 0, we have

H2((∪pi=1(ϕ1(Di) \ ∂Di)) ∩M \ (A \ ∂A)) < α (2.53)

for some isotopy {ϕs}s∈[0,1] ∈ Isout(U) (depending on α) which is identity on some

open neighborhood of (Σ \ Σ0) \ ∪pi=1(Di \ ∂Di).

Although all “small” disks with the same boundary are isotopic, which is crucial

in the proof of Theorem 2 in Meeks-Simon-Yau [35], not all of them are “outward iso-

topic”. However, in the next lemma, we see that it is almost true, modulo arbitrarily

small area.

Lemma 2.5.11. Let U and A be as in Theorem 2.5.10. Let Γ be a Jordan curve

in ∂A∩int(M) which is either closed or having endpoints on ∂M . Let F ⊂ ∂A be


a connected component of (∂A ∩ M) \ Γ, which is diffeomorphic to a disk, a half-

disk or an annulus. Let D ⊂ U ∩M be a genus 0 surface transversal to ∂M with

∂D \ ∂M = ∂F ∩ \∂M = Γ, and D ∩ F = ∅. In addition, we assume that F ∪ Dbounds a unique compact set K ⊂M modulo ∂M , i.e. ∂K∩int(M) = F ∪D.

Then, for any α > 0, there exists an isotopy {ϕs}s∈[0,1] ∈ Isout(U) supported on a

small neighborhood of K such that ϕs(x) = x for all x ∈ Γ and all s ∈ [0, 1], moreover,

H2((ϕ1(D) ∩M)∆F ) < α. (2.54)

In other words, we can outward isotope D to approximate F as close as we want.

Proof. We divide the situation into two cases according to whether the boundary

curve Γ is closed or not.

Case 1: Γ is a simple closed curve.

In this case, F is either a closed disk or a closed annulus in ∂A ∩M . In the

latter case, we will show that we can even find an isotopy {ϕs} supported on a

neighborhood of K, leaving Γ fixed, and ϕ1(D) ∩M = F .

After a change of coordinate, we can assume that

– U is the open ball of radius 2 in R3 centered at origin;

– A ⊂ U is the closed unit 3-ball centered at origin;

– M ∩ U = U ∩ {x3 ≥ 0} is the upper half-ball; and

– Γ = {(x1, x2, x3) ∈ R3 : x3 = 12, x2

1 + x22 = 3

4}.

First, we look at the case that F is a closed disk, i.e. F = {(x1, x2, x3) ∈ R3 :

x3 ≥ 12}. Let D be the genus 0 surface as given in the hypothesis. Note that

D meets ∂M at a finite number of simple closed curves Γi, i = 1, . . . , N , each

of which bounds a closed disk Di in ∂M ∩ U . Since D is a genus 0 surface

with boundary, it is clear that D ∪ F ∪ (D1 ∪ · · · ∪ DN) is homeomorphic to

a 2-sphere, and thus, the compact set K bounded by F ∪ D modulo ∂M is

homeomorphic to the unit 3-ball in R3. Observe that if D ∩ ∂M = ∅, then it


is trivial that we can isotope D to F , holding ∂M fixed. If D ∩ ∂M 6= ∅, we

then use a tangential isotopy to deform it so that it approximates a disk disjoint

from ∂M with boundary Γ, which in turn is isotopic to F . To see this, perturb

each closed disk Di into the interior of M such that the boundary of the disk

stays on D, call Di the perturbed disk, then there is a closed annulus Ai ⊂ D

such that ∂Ai = ∂Di ∪ ∂Di and Ai ∪ Di ∪ Di bounds a ball in K. Using a

tangential isotopy, we can deform D such that it agrees with (D \Ai)∪ Di with

an error in area as small as we want (one simply shrinks the size of the neck in

K). Repeat the whole procedure for each Di, one can deform D such that is

arbitrarily close to a disk disjoint from ∂M , and we are done.

In the case that F is a closed annulus, again let Γi be the set of simple closed

curves where D meets ∂M . Note that all except possibly one Γi bounds a

disk in K. For those which bounds a disk, we can repeat the “neck-shrinking”

argument as in the previous case to eliminate them. Therefore, we can assume

that Γi together with Γ0 = ∂F ∩ ∂M bounds a connected genus 0 surface Λ in

∂M . After a further change of coordinate, we can assume that K = Λ× [0, 12],

where we think of Λ as a subset of ∂M ⊂ R2. Next consider the outward

isotopy given by the vertical translations ϕs(x1, x2, x3) = (x1, x2, x3 − s) (with

some cutoff near ∂U so that it is supported in U), take a smooth function χ

on R2 such that χ = 0 along Γ0, χ = 1 outside a small neighborhood V of Γ0

disjoint from all Γi, and 0 < χ < 1 elsewhere. By choosing the neighborhood V

of Γ0 smaller and smaller, we see that the outward isotopy {χϕs} given by the

vertical translations with cutoff would deform D to approximate F as close as

we want. This implies our desired conclusion.

Case 2: Γ is an arc with endpoints on ∂M .

Assume the standard setting as before after a change of coordinate, suppose for

our convenience that Γ = {(x1, x2, x3) ∈ ∂A : x1 = 0}, and F = {(x1, x2, x3) ∈∂A : x1 ≥ 0}. Note that D intersects ∂M at a Jordan arc Γ1 with the

same endpoints as Γ and a (possibly empty) finite collection of disjoint simple

closed curves Γi, i = 2, . . . , N . Let Γ0 = F ∩ ∂M . By assumption, there is a


compact set K ⊂ M ∩ U such that ∂K = D ∪ F ∪ Λ where Λ is a genus 0

surface in ∂M with ∂Λ = ∪Ni=0Γi. For all those Γi, i = 2, . . . , N , which bounds

a disk in Λ, we can shrink down the neck as in Case 1. So we can assume

without loss of generality that Λ is connected. There are two further sub-cases:

either Γ0 ∪ Γ1 is the outermost boundary of Λ or it is not. In the first case,

similar to the second part of Case 1 above, we can assume (up to a change

of coordinate) that F = {(x1, x2, x3) ∈ R3 : x1 = 0, x3 ≥ 0, x22 + x2

3 ≤ 1},Λ = {(x1, x2, x3) ∈ R3 : x3 = 0, x1 ≥ 0, x2

1 + x22 ≤ 1} \ ∪Ni=2Di where Di is the

disk in ∂M bounded by the simple closed curve Γi, and D is the union of a

graph over Λ and n−1 cylinders contained in Γi× [0, 1]. Using a cutoff function

χ as before which is zero on Γ0 = F ∩ ∂M and the vertical translations, we

can deform D to approximate F as close as we want. Now we are left with the

case that Γ0 ∪Γ1 is not the outermost boundary of Λ. In this case, we can take

Λ = {(x1, x2, x3) ∈ R3 : x3 = 0, x21 + x2

2 ≤ 916} \ ∪Ni=2Di where Di is the disk in

∂M bounded by the simple closed curve Γi, and D is the union of a graph over

Λ and n− 1 cylinders contained in Γi × [0, 1]. Then a similar translation with

cutoff will deform D to approximate F and we are done.

Now we are ready to give a proof of Theorem 2.5.10.

Proof of Theorem 2.5.10. The proof is almost the same as that in Meeks-Simon-Yau

[35], except that not all small disks are outward isotopic. We will use Lemma 2.5.11

to get around this problem.

As in Meeks-Simon-Yau [35], we can assume that Σ0 = ∅. We proceed by induction

on q. Denote

(H)q Σ0 = ∅,∑q

j=1 H2(Fj) ≤ γ/8, E(Σ) ≤ γ/2 − 2

∑qj=1 H

2(Fj), and Σ is strongly

(γ, U)-irreducible, where γ = γ/4 + 4∑q

j=1 H2(Fj) + E(Σ).

(C)q the conclusion of Theorem 2.5.10 is true.

We will show that the statement “(H)q ⇒ (C)q” is true for all q. Assume it is true

for q − 1. We want to show by induction that it is true for q also.


Relabeling if necessary, we can assume that Fq is “innermost”, i.e. Fq ∩ Γj = ∅for all j 6= q. Since Σ is strongly (γ, U)-irreducible and H2(Fq) < γ. By Proposition

2.5.7, there exists a connected genus 0 surface D ⊂ Σ∩U∩M such that ∂D\∂M = Γq

and H2(D) < δ2/2. Since Fq is innermost, D ∩ Fq = ∅. Since both the area of Fq

and D are small, using Lemma 2.5.5, there is a unique compact set K ⊂ M which

is bounded by Fq ∪D modulo ∂M . Moreover, since F ∪D is a genus 0 surface, it is

easy to see that there exists a small neighborhood of K which is diffeomorphic to the

unit 3-ball whose boundary is disjoint from Σ∩M . Since we assume that Σ0 = ∅, we

know that the whole neighborhood is disjoint from (Σ ∩M) \D.

Replace D by Fq and write Σ∗ = (Σ \D)∪Fq (which is only a Lipschitz surface),

and Fq,ε = {x ∈ M : d(x, Fq) < ε}, for each ε > 0, we can select a continuous

tangential isotopy {ϕs}s∈[0,1] ∈ Istan(U) such that ϕs(Fq,ε) ⊂ Fq,ε, ϕs(x) = x for

x /∈ Fq,ε, furthermore,

H2((Σ∗ ∩ Fq,ε) ∩M) ≤ H2(ϕ1(Σ∗ ∩ Fq,ε) ∩M) ≤ H2((Σ∗ ∩ Fq,ε) ∩M) + ε (2.55)

and

ϕ1(Σ∗ ∩ Fq,ε) ∩ ∂A = ∅. (2.56)

In other words, we deform Σ∗ to detach Fq from ∂A. Let Σ∗ = ϕ1(Σ∗) (smooth by

suitably choosing ϕ1), for ε small enough, we have

(i) Σ∗ ∩ ∂A = ∪q−1j=1Γj,

(ii) H2((Σ∗∆Σ) ∩M) < H2(D) + H2(Fq) + ε,

(iii) H2(Σ∗ ∩M) < H2(Σ ∩M) + H2(Fq)−H2(D) + ε.

Notice (iii) implies

(iii)’ E(Σ∗) < E(Σ) + H2(Fq)−H2(D) + ε.

because Lemma 2.5.11 implies that

infΣ′∈JU (Σ)

H2(Σ′ ∩M) ≤ infΣ′∈JU (Σ∗)

H2(Σ′ ∩M). (2.57)


Taking ε < H2(Fq), following the same arguments in Meeks-Simon-Yau [35], we

see that Σ∗ satisfies (H)q−1, hence (C)q−1 holds by induction hypothesis. There must

be pairwise disjoint connected genus 0 surfaces ∆1, . . . , ∆p contained in Σ∗ ∩ U ∩Mwith ∂∆j \ ∂M ⊂ ∂A,

(∪pi=1∆i

)∩ (A \ ∂A) = Σ∗ ∩ (A \ ∂A) ∩M , and

p∑i=1

H2(∆i) ≤q−1∑j=1

H2(Fj) + E(Σ∗). (2.58)

Furthermore, for any α > 0,

H2(∪pi=1(Ψ1(∆i) \ ∂∆i) ∩M \ (A \ ∂A)) <α

2(2.59)

for some isotopy {Ψs}s∈[0,1] ∈ Isout(U) which fixes a neighborhood of (Σ∗ ∩ M) \∪pi=1(∆i \ ∂∆i). Reversing the isotopy ϕ used in (2.55) and (2.56), there are pairwise

disjoint connected genus 0 surfaces ∆1, . . . ,∆p ⊂ Σ∗ ∩M = ((Σ ∩M) \D) ∪ Fq with

(∪pi=1∆i) ∩ (A \ ∂A) = Σ∗ ∩ (A \ ∂A) ∩M , ∂∆i \ ∂M = ∂∆i \ ∂M , and

p∑i=1

H2(∆i) ≤q−1∑j=1

H2(Fj) + E(Σ) + H2(Fq)−H2(D), (2.60)

Furthermore,

H2(∪pi=1(Ψ1(∆i) \ ∂∆i) ∩M \ (A \ ∂A)) <3α

4(2.61)

for some isotopy {Ψs}s∈[0,1] ∈ Isout(U) which fixes a neighborhood of (Σ∗ ∩ M) \(∪pi=1(∆i \ ∂∆i) ∪ Fq,ε).

Recall that K is the compact set in U ∩M bounded by D∪Fq, by Lemma 2.5.11,

there exists a continuous isotopy {βs} ∈ Isout(U) supported on a neighborhood of K

fixing Γq = ∂D \ ∂M and

H2((β1(D)∆Fq) ∩M) <α

8. (2.62)

Moreover, we know that (Σ\D)∩K = ∅ because Σ0 = ∅. Consider the following two

cases: (i) Fq ⊂ ∪pi=1∆i; and (ii) Fq 6⊂ ∪pi=1∆i.


In case (i), if Fq ⊂ ∪pi=1∆i, by taking Dj0 = (∆j0 \ Fq) ∪D for the unique j0 such

that Fq ⊂ ∆j0 , and we select Dj = ∆j for all j 6= j0. Also, we define a continuous

outward isotopy ϕ = {ϕs} by ϕ = Ψ∗β; by smoothing ϕ we obtain an outward isotopy

ϕ satisfying the required conditions. Here Ψ ∗ β is defined by Ψ ∗ βs(x) = β2s(x) if

0 ≤ s ≤ 12, and Ψ ∗ βs(x) = Ψ2s−1(β1(x)) if 1

2< s ≤ 1.

In case (ii), if Fq 6⊂ ∪pi=1∆i, we define the set of pairwise disjoint connected genus

0 surfaces D1, . . . , Dp+1 by setting Dj = ∆j, j = 1, . . . , p, and Dp+1 = D. In this

case, we define a continuous isotopy ϕ by setting ϕ = β ∗ (Ψ ∗ β), where β = {βs} is

a smooth outward isotopy such that βs(x) = x for all x ∈ (Σ∩M) \D and s ∈ [0, 1],

and such that β1(Fq)∩M is a genus 0 surface D ⊂ A∩M with ∂D \∂M = ∂D \∂M ,

D ∩ ∂A = ∂D ∩ ∂A, and D ∩ Ψs(Σ∗) = Γq for all s ∈ [0, 1]. Such a β exists once we

show the claim that in case (ii), there is a neighborhood W of Γq = ∂D\∂M such that

W∩D ⊂ A. Otherwise, we would have W with Γq ⊂ W and W∩(Σ\D)∩M ⊂ A\∂A,

and this would imply that Fq ⊂ ∪pi=1∆i since (∪pi=1∆i)∩ (A\∂A) = Σ∗∩ (A\∂A)∩M(see the statement above (2.60)), thus contradicting we are in case (ii). By smoothing

ϕ we then again obtain the required outward isotopy ϕ.

In each of the above cases, we have, by (2.60), that

p∑i=1

H2(∆i) ≤q−1∑j=1

H2(Fj) + E(Σ) + H2(Fq)−H2(D),

and hence

p∑i=1

H2(Di) ≤q−1∑j=1

H2(Fj) + E(Σ) + H2(Fq)−H2(D) + H2(D)

=

q∑j=1

H2(Fj) + E(Σ)

This proves that statement (C)q and the proof is finished by induction.

We are going to need a replacement lemma about finite collection of genus 0

surfaces with disjoint boundaries.


Lemma 2.5.12. Let A ⊂ M be a closed subset which is diffeomorphic to the unit

3-ball such that A∩ ∂M is diffeomorphic to the closed unit disk. Suppose D1, . . . , DR

are connected genus 0 surfaces in A∩M with Di \∂Di ⊂ A \∂A and ∂Di ⊂ ∂A∩M .

Also, assume that (∂Di \ ∂M) ∩ (∂Dj \ ∂M) = ∅ and that either Di ∩Dj = ∅ or Di

intersects Dj transversally for all i 6= j.

Then, there exists pairwise disjoint connected genus 0 surfaces D1, . . . , DR in A∩M with Di \ ∂Di ⊂ A \ ∂A, ∂Di ∩ ∂A = ∂Di ∩ ∂A and H2(Di) ≤ H2(Di) for

i = 1, . . . , R.

Proof. Assume that R ≥ 2 and that D1, . . . , DR−1 are already pairwise disjoint. If we

can prove the required result in this case, then the general case follows by induction

on R.

Let Γ1, . . . ,Γq be pairwise disjoint Jordan curves (either closed or have boundaries

on ∂M), not necessarily connected, such that

DR ∩(∪R−1i=1 Di

)= ∪qj=1Γj, (2.63)

also, for each j = 1, . . . , R, Γj divides each Di which contains Γj into two genus 0

surfaces (maybe disconnected) Di \ Γj = D′i ∪D′′i with ∂D′i \ ∂M = Γj = ∂D′′i \ ∂M(since ∂Di ∩M are pairwise disjoint by assumption, either D′i or D′′i is disjoint from

∂Di\∂M) and, as an inductive hypothesis, assume the lemma is true whenever (2.63)

holds with r ≤ q − 1 on the right hand side (with D1, . . . , DR−1 still being assumed

pairwise disjoint).

For each j = 1, . . . , q, let Ej be the part of DR\Γj which is disjoint from ∂DR\∂M .

Hence, ∂Ej \ ∂M = Γj. Let Fj be the corresponding part in ∪R−1i=1 Di \ Γj which is

disjoint from ∪q−1i=1∂Di \ ∂M . Hence, ∂Fj \ ∂M = Γj. Let K ⊂ ∪Ri=1Di be a genus 0

surface with ∂K \ ∂M = Γj0 for some j0 such that

H2(K) ≤ minj=1,...,q

{H2(Ej),H2(Fj)}. (2.64)

Let J 6= K be the other genus 0 surface in ∪Ri=1Di such that ∂J\∂M = ∂K\∂M = Γj0 .


Evidently we must have

(K \ ∂K) ∩ (∪i 6=i0Di) = ∅, (2.65)

where i0 is such that K ⊂ Di0 . Let i1 6= i0 be such that J ⊂ Di1 (note that then one of

i0, i1 is equal to R), and define Dj = Dj if j 6= j1 and Di1 = (Di1\J)∪K. By (2.65) we

have that each Dj is an embedded genus 0 surface, and clearly ∂Dj \∂M = ∂Dj \∂M ,

H2(Dj) ≤ H2(Dj), D1, . . . , DR are pairwise disjoint and

DR ∩(∪R−1i=1 Di

)= K ∪ (∪j 6=j0Γj) . (2.66)

By smoothing Di1 near Γj0 and making a slight perturbation near K, we then ob-

tain genus 0 surfaces D∗1, . . . , D∗R with ∂D∗j \ ∂M = ∂Dj \ ∂M , H2(D∗j ) ≤ H2(Dj),

D∗1, . . . , D∗R−1 pairwise disjoint, and (using (2.66)),

D∗R ∩(∪R−1j=1 D

∗j

)= ∪j 6=j0Γj.

Hence, we can apply the inductive hypothesis to the collection {D∗j}, thus obtaining

the required collection D1, . . . , DR.

2.5.2 Minimizing sequence of genus 0 surfaces

In this section, we recall a result by Jost [30] on regularity for the minimization

problem for genus 0 surfaces with partially free boundary.

Let A ⊂ M be an admissible open set. Let Γ ⊂ ∂A∩M be an embedded smooth

curve in M which either meets ∂M at the two endpoints transversally or is disjoint

from ∂M . Let M = M(0,Γ) be the set of all genus 0 surfaces D contained in M with

Γ as boundary modulo ∂M , i.e. ∂D \ ∂M = Γ, and which meets ∂M transversally.

We say that Dk is a minimizing sequence for M if

H2(Dk) ≤ infD∈M

H2(D) + εk (2.67)

for some positive real numbers εk → 0 as k →∞.


Theorem 2.5.13 (Jost). Using the notation above, let Dk ∈ M be a minimizing

sequence for M, and suppose Dk converges to V in the sense of varifolds in M . Then

for each point x0 ∈supp‖V ‖ ∩ ∂M , there are n ∈ N, ρ > 0 (both depending on x0)

and an embedded minimal surface Γ in M meeting ∂M orthogonally with

V xBρ(x0)= nv(Γ)

where v(Γ) is the varifold represented by Γ with multiplicity one.

2.5.3 Convergence of the minimizing sequence

In this section, we prove the main regularity result, which we recall here:

Theorem 2.5.14. Let U ⊂ M be an admissible open set, suppose {Σk} is a minimiz-

ing sequence for the minimization problem (Σ, Isout(U)) defined in Definition 2.5.2

so that

(i) Σk intersects ∂M transversally for each k, and

(ii) Σk ∩M converge weakly to a varifold V ∈ V(M).

Then,

(a) V = Γ, for some compact embedded minimal surface Γ ⊂ U ∩M with smooth

boundary (except possibly at ∂U ∩ ∂M) contained in ∂(U ∩M);

(b) The fixed boundary of Γ is the same as Σ ∩ (∂U ∩M);

(c) Γ meets ∂M ∩ U orthogonally along the free boundary of ∂Γ;

(d) Γ is stable with respect to all tangential variations supported in U .

Proof. Let {Σk} be a minimizing sequence for the minimization problem (Σ, Isout(U))

with Σk ∩M converging weakly to a varifold V in M and Σk intersects ∂M transver-

sally for each k. Using the same argument as in Meeks-Simon-Yau, we assume that

(Σk)0 = ∅ (see the paragraph above Theorem 2.5.10 for definition) for all k and Σk


is strongly (γ, U)-irreducible for all sufficiently large k, for some fixed 0 < γ < δ2/9.

Furthermore, we have

H2(Σk ∩M) ≤ infΣ∈JU (Σk)

H2(Σ ∩M) + εk (2.68)

where εk → 0 as k →∞.

As noted before, interior regularity and regularity at fixed boundary have been

discussed in Meeks-Simon-Yau [35] and De Lellis-Pellandini [13]. So we only have to

prove regularity at free boundary.

Let x0 ∈supp(‖V ‖) ∩ U ∩ ∂M , and ν0 be the outward unit normal at x0 ∈ ∂M .

Define x1 = expx0(εν0) be a point outside M which is very close to x0 (by choosing

ε very small). Let ρ0 > 0 be chosen small so that all the geodesic balls Bρ(x1) in M

are admissible open sets in the sense of Definition 2.5.1 for all 0 < ρ ≤ ρ0. Note that

we have to move the center of the balls from x0 to x1 in order for (v) of Definition

2.5.1 to hold.

First of all, we want to show that V is tangentially stationary in U . Let X ∈C∞tan(M, TM) be supported in U , and {ϕs}s∈(−ε,ε) be the isotopy generated by X. By

(2.68),

H2(Σk ∩M) ≤ H2(ϕs(Σk) ∩M) + εk (2.69)

for all k. Note that ϕs(Σk)∩M = ϕs(Σk ∩M) since {ϕs} ∈ Istan(U), take k →∞ in

(2.69), we get

‖V ‖(M) ≤ ‖(ϕs)]V ‖(M) (2.70)

for all s ∈ (−ε, ε). This shows that V is tangentially stable, so V is of course tangen-

tially stationary. Therefore, we can apply the monotonicity formula (Theorem 2.1.5)

to V .

By the coarea formula, we have∫ ρ

ρ−σH1(Σk ∩ ∂Bs(x1) ∩M) ds ≤ H2(Σk ∩ (Bρ(x1) \Bρ−σ(x1)) ∩M) (2.71)

for almost every ρ ∈ (0, ρ0) and every σ ∈ (0, ρ), where Bs(x1) is the closed geodesic


ball in M of radius s centered at x1. Taking σ = ρ/2, Theorem 2.1.5 then gives∫ ρ

ρ/2

H1(Σk ∩ ∂Bs(x1) ∩M) ds ≤ cρ2 (2.72)

for all sufficiently large k, where c depends only on M and M and any upper bound for

ρ−20 (‖V ‖(Bρ0(x1)) + ‖V ‖(Bρ0(x1))) (recall that B is the reflection of B across ∂M).

Hence, we can find a sequence {ρk} ⊂ (3ρ/4, ρ) such that Σk intersects ∂Bρk(x1)

transversally and such that

H1(Σk ∩ ∂Bρk(x1) ∩M) ≤ cρ ≤ cηρ0 (2.73)

for all sufficiently large k, provided ρ ≤ ηρ0, where for the moment η ∈ (0, 1) is arbi-

trary. If η is sufficiently small, we see from (2.73) that Theorem 2.5.10 is applicable.

Hence, there are connected genus 0 surfaces D(1)k , . . . , D

(qk)k ⊂ Σk ∩M and for any

α > 0, isotopies {ϕ(k)t }t∈[0,1] ∈ Isout(Bρ0(x1)) such that

∂D(j)k \ ∂M ⊂ ∂Bρk(x1), (2.74)

Σk ∩Bρk(x1) ∩M =(∪qkj=1D

(j)k

)∩Bρk(x1), (2.75)

H2(∪qkj=1(ϕ(k)1 (D

(j)k ) \ ∂D(j)

k ) ∩M \ (Bρk(x1) \ ∂Bρk(x1))) < α, (2.76)

andqk∑j=1

H2(D(j)k ) ≤ cρ2 ≤ cη2ρ2

0, (2.77)

where c is independent of k, η and ρ. Since H2(D(j)k ) ≤ cη2ρ2

0, we know that for η

sufficiently small, by the modified replacement lemma with free boundary (see Lemma

4.4 in Jost [30]), there are connected genus 0 surfaces D(j)k contained in M with

∂D(j)k \ ∂M = ∂D

(j)k \ ∂M, D

(j)k \ ∂D

(j)k ⊂ Bρk(x1), (2.78)

and

H2(D(j)k ) ≤ H2(D

(j)k ). (2.79)


Combining (2.68) and (2.79), using Lemma 2.5.12, (2.74), (2.76) and (2.78), D(j)k is

a minimizing sequence among all genus 0 surfaces with fixed boundary ∂D(j)k with

any number of free boundaries on ∂M . By Theorem 2.5.13, we know that for each

x0 ∈supp(V ) ∩ ∂M , there exist n ∈ N and ρ > 0 and an embedded minimal surface

Σ meeting ∂M orthongonally with V = nΣ on Bρ(x0). This finishes the proof of

Theorem 2.5.14.

2.6 Regularity of almost minimizing varifolds

In this section, we define the notion of “good replacement property” for tangentially

stationary varifolds and prove that if there exists sufficiently many replacements, then

the varifold must be a smooth minimal surface with free boundary. In the second

half, we will see how to construct these replacements if the tangentially stationary

varifold is the limit of an almost minimizing min-max sequence as in Theorem 2.4.6.

Definition 2.6.1. Let V ∈ V(M) be a tangentially stationary varifold and U ⊂ N

be an open subset. We say that V ′ ∈ V(M) is a replacement for V in U if and only if

1. V ′ is tangentially stationary;

2. V ′ = V on G(M \ U) and ‖V ′‖(M) = ‖V ‖(M);

3. V ′x(U∩M) is (an integer multiple of) a smooth stable embedded (not necessarily

connected) minimal surface Σ ⊂ M meeting ∂M orthogonally. Here, “sta-

ble” means that the second variation is nonnegative with respect to variations

{ϕt}t∈(−ε,ε) supported in U and ϕt(M) = M for all t.

Definition 2.6.2. Let V ∈ V(M) be tangentially stationary and U ⊂ N be an open

subset. We say that V has the good replacement property in U if and only if all the

following hold.

(a) There is a positive function r1 : U → R such that for every annulus An1 ∈ANr(x)(x), there is a replacement V ′ for V in An1 such that (b) holds;

(b) There is another positive function r2 : U → R such that


(i) V ′ has a replacement V ′′ in any An2 ∈ ANr(x)(x), for the same x and r as

in (a), and V ′′ satisfies (c) below;

(ii) V ′ has a replacement in any An∈ ANr2(y)(y) for any y ∈ U .

(c) There is yet another positive function r3 : U → R such that V ′′ has a replace-

ment in any An∈ ANr3(y)(y) for any y ∈ U .

The key result in this section is the following regularity theorem.

Theorem 2.6.3. If V has the good replacement property in an open set U ⊂ N , then

V is a smooth embedded minimal surface in U∩int(M) with smooth free boundary on

U ∩ ∂M .

The proof of interior regularity can be found in Colding-De Lellis [9]. Therefore,

we will only focus on regularity on the free boundary. To prove Theorem 2.6.3, we

first state a generalization of two lemmas from Colding-De Lellis [9] adapted to the

free boundary setting.

Lemma 2.6.4 (maximum principle for tangentially stationary varifolds). Let U be

an open set of M and W ∈ V(U) be tangentially stationary. If K ⊂⊂ U is a smooth

strictly convex set and x ∈supp(‖W‖) ∩ ∂K, then for every r > 0,

(Br(x) \K) ∩ supp(‖W‖) 6= ∅. (2.80)

Proof. See Theorem 1 of White [55]. Actually the conclusion holds even when ∂U is

only strictly mean convex with respect to the inward unit normal.

Next we recall a lemma from Colding-De Lellis [9].

Lemma 2.6.5 (Generic transversality). Let x ∈ M , and V ∈ V(M) be a tangentially

stationary integer rectifiable varifold. Assume T is the subset of supp(‖V ‖) where

T = {y ∈ supp(‖V ‖) : T (y, V ) consists of a plane transversal to ∂Bd(x,y)(x)}.(2.81)

If ρ is less than the injectivity radius inj(M) of M , then T is dense in supp(‖V ‖) ∩Bρ(x) \ ∂M .


The next lemma tells us what we can say about the tangentially stationary varifold

if it can be replaced once.

Lemma 2.6.6. Let U ⊂ N be open and V ∈ V(M) be a tangentially stationary

varifold in U . If there exists a positive function r : U → R such that V has a

replacement in any annulus An∈ ANr(x)(x), then V is integer rectifiable in U ∩Mand if x ∈supp(‖V ‖) ∩ (int(M) ∩ U), then θ(x, V ) ≥ 1 and any tangent cone to V

at x is an integer multiple of a plane. Moreover, if x ∈supp(V ) ∩ (∂M ∩ U), then

θ(x, V ) ≥ 12

and any tangent cone to V at x is an integer multiple of a half-plane

orthogonal to Tx∂M .

Proof. Fix an x ∈ supp(‖V ‖) ∩ (∂M ∩ U). Since V is tangentially stationary, the

monotonicity formula for free boundary (Theorem 2.1.5) gives r > 0 and a constant

C > 0 (depending only on M and M) such that for all y in some neighborhood of

∂M in M ,

‖V ‖(Bσ(y)) + ‖V ‖(Bσ(y))

σ2≤ C‖V ‖(Bρ(y)) + ‖V ‖(Bρ(y))

ρ2for all 0 < σ < ρ < r

(2.82)

Choose ρ < r(x)/2 and so that 4ρ < r is smaller than the convexity radius of M and

ρ < d(x, ∂U). Since 2ρ < r(x), there is a replacement V ′ ∈ V(M) for V in the annulus

An(x, ρ, 2ρ). First of all, V ′ 6= 0 on An(x, ρ, 2ρ). Otherwise, since V = V ′ in Bρ(x),

we have x ∈supp(‖V ′‖) and there would be a σ ≤ ρ such that V ′ touches ∂Bσ(x)

from the interior, i.e. σ = maxy∈supp(‖V ′‖) d(y, x). However, since Bσ(x) is convex,

this would contradict Lemma 2.6.4. Therefore, V ′ is a non-empty smooth surface in

An(x, ρ, 2ρ) which meets ∂M orthogonally, and so there is some y ∈An(x, ρ, 2ρ)\∂Mwith θ(V ′, y) ≥ 1. By the monotonicity formula (2.82), and notice that y /∈ ∂M ,

‖V ‖(B4ρ(x))

16ρ2=‖V ′‖(B4ρ(x))

16ρ2≥ ‖V

′‖(B2ρ(y))

16ρ2≥ π

4C. (2.83)

For x ∈supp(‖V ‖)∩int(M)∩U , the usual monotonicity formula for stationary varifold

gives a similar lower bound. Hence, θ(x, V ) is bounded uniformly from below on

supp(‖V ‖), applying the rectifiability theorem, we conclude that V is rectifiable.


The interior case has been discussed in Colding-De Lellis [9], and we know that V

is integer rectifiable. So it remains to prove the free boundary case in the theorem.

Fix x ∈supp(‖V ‖)∩ (∂M ∩U), and a sequence ρn ↓ 0 such that V xρn converges weakly

to a tangent cone C ∈ TV (x, V ) which is stationary with respect to all variations

tangential to Tx∂M . By a change of coordinate, we can assume that Tx∂M has inward

pointing normal (0, 0, 1). We will show that C is an integer multiple of a half plane

H. Since C is tangentially stationary, H must be orthogonal to Tx∂M , hence contain

(0, 0, 1).

First, we place V by V ′n in the annulus An(x, ρn/4, 3ρn/4) and set W ′n = (T xρn)]V

′n.

After possibly passing to a subsequence, we can assume that W ′n → C ′ weakly, where

C ′ is a stationary varifold with respect to tangential variations. By the definition of

replacements, we have

C ′ = C in B 14∪ An(0,

3

4, 1), (2.84)

and

‖C ′‖(Bρ) = ‖C‖(Bρ) for ρ ∈ (0,1

4) ∪ (

3

4, 1), (2.85)

where Bs is the ball the radius s in R3 centered at the origin. Since C is a cone, using

(2.85), we have

‖C ′‖(Bσ)

σ2=‖C ′‖(Bρ)

ρ2for all σ, ρ ∈ (0,

1

4) ∪ (

3

4, 1). (2.86)

Hence, the stationarity of C ′ and the monotonicity formula imply that C ′ is also

a cone. By Theorem 2.1.7, W ′n converges to a stable embedded minimal surface in

An(x, 1/4, 3/4), with respect to variation fields in C∞tan(M, TM). This means that

C ′xAn(x, 1/4, 3/4) is an embedded minimal cone in the classical sense and hence is

supported on a half disk containing the origin. The minimal cone is not the x-y plane

since each W ′n meets ∂M orthogonally. This forces C ′ and C to coincide and be an

integer multiple of the same half plane perpendicular to Tx∂M .

We now return to the proof of the main result (Theorem 2.6.3) of this section.


Proof of Theorem 2.6.3. The interior regularity is covered in Colding-De Lellis [9], so

we only give a proof for free boundary regularity here.

Fix x ∈supp(‖V ‖) ∩ ∂M ∩ U . Choose ρ small such that ρ < r(x)/2 and 2ρ is

less than the convexity radius of M , by the good replacement property (a), we can

find a replacement V ′ for V in the annulus An(x, ρ, 2ρ). Let Σ′ be the stable minimal

surface given by V ′ in An(x, ρ, 2ρ). For any t ∈ (ρ, 2ρ) and s ∈ (0, ρ), by the good

replacement property (b)(i), we can find a replacement V ′′ of V ′ in An(x, s, t). Let

Σ′′ be the stable minimal surface given by V ′′ in An(x, s, t).

First, we choose some t ∈ (ρ, 2ρ) such that Σ′ intersects ∂Bt(x) transversally.

Such a t exists because Σ′ is a smooth surface in the annulus An(x, ρ, 2ρ). Next, we

show that Σ′∩An(x, t, 2ρ) can be glued to Σ′′ ⊂An(x, s, t) smoothly. It has already

been shown that they glue together smooth in the interior in Colding-De Lellis [9],

so it suffices to show that they also glue together smoothly along the free boundary.

Fix a point y ∈ Σ′ ∩ ∂Bt(x) ∩ ∂M , and a sufficiently small radius r so that

Σ′ ∩Br(y) is a half-disk orthogonal to ∂M and γ = Σ′ ∩ ∂Bt(x) ∩Br(y) is a smooth

arc perpendicular to ∂M . As in Colding-De Lellis [9], we can assume, without loss of

generality, that Br(y) is the unit ball B in R3 centered at origin, ∂M ∩Br(y) = {z2 =

0}∩B, M∩Br(y) = {z2 ≥ 0}∩B. Moreover, ∂Bt(x)∩Br(y) = {z1 = 0}∩B. Suppose

Σ′∩Br(y) is the graph of a smooth function g(z1, z2) defined on {z2 ≥ 0, z3 = 0}∩B.

Hence, γ = {(0, z2, g(0, z2) : z2 ≥ 0}.The replacement V ′′ consists of Σ′′ ∪ (Σ′ \ Bt(x)) in Br(y). By Lemma 2.6.6,

using the fact that V ′′ satisfies (c) in Definition 2.6.2, T (y, V ′′) consists of a family

of (integer multiples) of half-planes orthogonal to {z1 = 0} ∩B (in other words, they

contain the vector (0, 1, 0)). Since Σ′ is regular and transversal to {z1 = 0}, each

half plane P ∈ T (y, V ′′) coincides with the half plane TyΣ′ in {z1 < 0}. Therefore,

T (y, V ′′) = {TyΣ′}. Now,following the argument in Colding-De Lellis [9], we obtain

a function g′′(z1, z2) ∈ C1({z1 ≥ 0, z2 ≥ 0}) such that

Σ′′ ∩Br(y) = {(z1, z2, g′′(z1, z2)) : z1 > 0, z2 ≥ 0} (2.87)

g′′(0, z2) = g′(0, z2) and ∇g′′(0, z2) = ∇g′(0, z2) for all z2 ≥ 0. (2.88)


Since Σ′ and Σ′′ meets ∂M orthogonally, we have the free boundary condition∇g′(z1, 0) =

(0, 0) for all z1 ≤ 0 and ∇g′′(z1, 0) = (0, 0) for all z1 > 0. By the reflection principle,

one obtain a continuous function G defined on the unit disk D = {z3 = 0} ∩ B such

that G is smooth and satisfies the minimal surface equation on the punctured disk

D \ 0. Hence by standard interior regularity for second order uniformly elliptic PDE,

G is smooth across the origin.

By the maximum principle Lemma 2.6.4, we have shown that for any s < ρ, Σ′ can

be extended to a surface Σs in An(x, s, 2ρ) such that if s1 < s2 < ρ, then Σs1 = Σs2

in An(x, s2, 2ρ). Thus, Σ = ∪0<s<ρΣs is a stable minimal surface with free boundary

on ∂M and Σ \Σ ⊂ ∂B2ρ(x)∪ ∂M ∪ {x}. Next, we show that V coincides with Σ in

Bρ(x) \ {x}. Recall that V = V ′ in Bρ(x). Fix any y ∈ supp(‖V ‖)∩Bρ(x) \ {x} and

set s = d(x, y). Since Σ meets ∂M orthogonally, H2(Σ∩ ∂M) = 0, so we can assume

y ∈int(M) and T (y, V ) consists of a multiple of a plane π transversal to ∂Bs(x) (by

Lemma 2.6.5), then we know that y ∈ Σ as in Colding-De Lellis [9]. Therefore, (2)

in the definition of replacement implies that V = Σ on Bρ(x).

It remains to show that x is a removable singularity for Σ. By Lemma 2.6.6, every

C ∈ T (x, V ) is a multiple of a half-plane orthogonal to Tx∂M . Following Colding-De

Lellis [9], for ρ sufficiently small, there exists natural numbers N(ρ) and mi(ρ) such

that

Σ ∩ An(x, ρ/2, ρ) = ∪N(ρ)i=1 mi(ρ)Σi

ρ (2.89)

where each Σiρ is a Lipschitz graph over a planar half-annulus, with the Lipschitz

constants uniformly bounded independent of ρ. Hence, we get N minimal punctured

half-disks Σi with

Σ ∩Bρ(x) \ {x} = ∪Ni=1miΣi. (2.90)

By Allard regularity (Theorem 2.1.6), we see that x is a removable singularity for

each Σi. Finally, by the Hopf boundary lemma for uniformly elliptic second order

PDE, N must be one. This completes the proof.

To finish the proof of Theorem 2.2.7, it remains to construct replacements for

limits of almost minimizing min-max sequences. We will need the regularity result

proved in section 2.5.


Let V be as in Theorem 2.4.6 and fix an annulus An∈ ANr(x)(x). Set

Isj(An) = {{ϕs} ∈ Isout(An) : H2(ϕs(Σj) ∩M) ≤ H2(Σj ∩M) +

1

8j∀s ∈ [0, 1]}

The argument is complete once we proved the following lemma.

Lemma 2.6.7. For each j, suppose we have a minimizing sequence {Σj,k}k∈N for the

problem (Σj, Isj(An)) that converges weakly to a varifold V j.

Then, Vj is a stable minimal surface in An with free boundary on ∂M . Moreover,

any V ∗ which is the limit of a subsequence of {V j} is a replacement for V (in the

sense of Definition 2.6.1).

Proof. The proof of the second assertion is exactly as that in Proposition 7.5 in

Colding-De Lellis [9]. So we only prove the first assertion here.

Without loss of generality, we assume that j = 1, and we write V ′, Σk and Σ

in place of V j, Σj,k and Σj respectively. Clearly V ′ is stationary and stable in An,

by its minimizing property. Thus, we only need to prove regularity. The proof

follows exactly as in Colding-De Lellis [9] except that we are using Theorem 2.5.14

and the following lemma instead. (Also, we use the compactness for stable minimal

surfaces with free boundary (Theorem 2.1.7) instead of the usual compactness for

stable minimal surfaces.)

Lemma 2.6.8. Let x ∈ An, and assume that {Σk} is minimizing for the problem

(Σ, Is1(An)). Then, there exists ε > 0 such that for k sufficiently large, the following

holds:

(Cl) For any {ϕs} ∈ Isout(Bε(x)) with H2(ϕ1(Σk)∩M) ≤ H2(Σk ∩M), there exists

another isotopy {φs} ∈ Isout(Bε(x)) such that ϕ1 = φ1 and

H2(φs(Σk) ∩M) ≤ H2(Σk ∩M) +

1

8for all s ∈ [0, 1].

Moreover, ε can be chosen so that (Cl) holds for any sequence {Σk} which is mini-

mizing for the problem (Σ, Is1(An)) and with Σj = Σj on M \Bε(x).


The proof of Lemma 2.6.8 is a rescaling argument. Exactly the same argument

(using the corresponding results in the free boundary case) works here. So we are

done.

2.7 Genus bound

In this section, we observe that a result in De Lellis-Pellandini [13] which controls the

topological type of the minimal surface constructed by min-max arguments continue

to hold in the case of free boundary. (We will assume that all surfaces in a sweep-out

is orientable as in De Lellis-Pellandini [13].) The proof is exactly the same as in De

Lellis-Pellandini [13]. One only has to note that a compact smooth surface Γ has

genus g if and only if the image of the map r : H1(Γ;Z)→ H1(Γ, ∂Γ;Z) is Z2g when

Γ is orientable, or the image is Zg−1 × Z2 if Γ is non-orientable. The lifting lemma

(Proposition 2.1 in De Lellis-Pellandini [13]) is still valid and hence the proof goes

through.

Theorem 2.7.1. Let Σj = Σjtj ∩M be a sequence which is almost minimizing in suffi-

ciently small annuli which intersects ∂M transversally for all j and V be the varifold

limit of Σj as j → ∞. Write V =∑N

i=1 niΓi where Γi are connected components of

Σ, counted without multiplicity and ni are positive. Let O be the set of those Γi which

are orientable and N be those which are non-orientable. Then

∑Γi∈O

g(Γi) +1

2

∑Γi∈N

(g(Γi)− 1) ≤ g0 = lim infj↑∞

lim infτ→tj

g(Σjτ ). (2.91)

where g(Γ) denotes the genus of a smooth compact surface Γ (possibly with boundary).

On the other hand, we note that it is impossible to get a similar bound on the

connectivity (i.e. number of free boundary components) of the minimal surface. A

direct corollary of Theorem 2.7.1 and Theorem 2.0.1 is the following. Noting that

there is no closed minimal surface in R3.

Corollary 2.7.2. Any smooth compact domain in R3 contains a non-trivial embed-

ded minimal surface Σ with non-empty boundary which is a free boundary solution


and such that

(i) either Σ is an orientable genus zero surface, i.e. a disk with holes;

(ii) or Σ is a non-orientable genus one surface, i.e. a Mobius band with holes.

This follows from the observation that any such domain can be swept out by

surfaces with genus zero. In fact, we can generalize the result to arbitrary 3-manifold.

Recall that for any orientable closed 3-manifold M , the Heegaard genus of M is the

smallest integer g such that M = Σ1 ∪ Σ2, where Σ1 ∩ Σ2 is an orientable surface

of genus g and each Σi, i = 1, 2, is a handlebody of genus g. For manifolds with

boundary, we make the following definition.

Definition 2.7.3. Let M be a compact 3-manifold with boundary. We define the fill-

ing genus of M to be the smallest integer g such that there exists a smooth embedding

of M into a closed orientable 3-manifold M with Heegaard genus g.

Since any closed 3-manifold with Heegaard genus g has a non-trivial sweep-out by

surfaces with genus as most g. The min-max construction on the saturation of such

a sweep-out together with the genus bound (Theorem 2.7.1) above give the following

result.

Corollary 2.7.4. Any smooth compact orientable 3-manifold M with boundary ∂M

with filling genus g contains a nonempty embedded smooth minimal surface Σ with

free boundary and the genus of Σ is at most g if it is orientable; and at most 2g + 1

if it is not orientable.

Chapter 3

Instability results for minimal

surfaces in manifolds with positive

isotropic curvature

In this chapter, we study complete minimal surfaces Σ in an n-dimensional Rieman-

nian manifold M (n ≥ 4) which minimize area up to second order. In particular,

we prove that if n = 4 and the ambient manifold M is orientable and has uniformly

positive isotropic curvature, then there does not exist a complete stable minimal sur-

face which is uniformly conformally equivalent to the complex plane C. Note that we

do not require M to be compact. Therefore the result applies to universal covers of

compact manifolds with positive isotropic curvature. At the end of the chapter, we

prove that the same result holds for any n ≥ 4 but M satisfies the stronger condition

that it has uniformly positive complex sectional curvature, where M need not be

orientable.

The main theorem in this chapter is the following:

Theorem 3.0.5 (Main Theorem B). Let M be a 4-dimensional complete orientable

Riemannian manifold with uniformly positive isotropic curvature. Let C be the com-

plex plane equipped with the standard flat metric. Then there does not exist a stable

immersed minimal surface Σ in M which is uniformly conformally equivalent to C.

70

CHAPTER 3. MINIMAL SURFACES IN PIC MANIFOLDS 71

An outline of this chapter is as follows. We begin with some definitions and

background in section 3.1. In section 3.2, we prove a vanishing theorem for the ∂-

operator on holomorphic bundles satisfying an “eigenvalue condition”. In section 3.3,

we describe Hormander’s weighted L2-method and use it to construct holomorphic

sections with controlled growth, assuming the “eigenvalue condition”. In section 3.4,

we apply the results to prove Main Theorem B. In section 3.5, we prove a density

lemma used in section 3.3. Finally, we discuss some open questions and conjectures.

3.1 Definitions and preliminaries

For basic definitions about manifolds with positive isotropic curvature, one can refer

to section 1.3. To state the main result in this paper, we need the following definition.

Definition 3.1.1. A Riemannian surface (Σ, h) is uniformly conformally equivalent

to the complex plane C if there is a (conformal) diffeomorphism φ : C→ Σ, a positive

smooth function λ on C and a constant C > 0 such that

φ∗h = λ2|dz|2 with1

C≤ λ2.

Our theorem states that surfaces of this type cannot arise as stable minimal sur-

faces in any orientable 4-manifold with uniformly positive isotropic curvature.

Theorem 3.1.2. Let M be a 4-dimensional complete orientable Riemannian manifold

with uniformly positive isotropic curvature. Let C be the complex plane equipped with

the standard flat metric. Then there does not exist a stable immersed minimal surface

Σ in M which is uniformly conformally equivalent to C.

If we assume that M has uniformly positive complex sectional curvature, then the

result holds in any dimension, and without the orientability assumption on M .

Theorem 3.1.3. Let M be an n-dimensional complete (not necessarily orientable)

Riemannian manifold with uniformly positive complex sectional curvature. Let C be

the complex plane equipped with the standard flat metric. Then there does not exist a


stable immersed minimal surface Σ in M which is uniformly conformally equivalent

to C.

3.2 A vanishing theorem

Throughout this chapter, C will denote the standard complex plane with the flat

metric

ds2 = dx2 + dy2 = |dz|2.

Suppose E is a holomorphic vector bundle over C with a compatible Hermitian metric.

Let ∂ denote the ∂-operator associated to E. Assume that (E, ∂) satisfies the following

“eigenvalue condition”: these exists some positive constant κ0 > 0, and a sufficiently

small constant ε0 > 0 (depending on κ0) such that for all 0 < ε < ε0, we have

κ0

∫C|s|2e−ε|z| dxdy ≤

∫C|∂s|2e−ε|z| dxdy (3.1)

for all compactly supported smooth sections s of E, we write s ∈ C∞c (E).

For any positive continuous function ϕ on C, we can define an L2-norm on C∞c (E)

by

‖s‖L2(E,ϕ) =

(∫C|s|2ϕ dxdy

) 12

.

Let L2(E,ϕ) be the Hilbert space completion of C∞c (E) with respect to the weighted

L2-norm ‖ · ‖L2(E,ϕ). In other words, s ∈ L2(E,ϕ) if and only if s is a measurable

section of E with ‖s‖L2(E,ϕ) <∞.

In this section, we prove a vanishing theorem for the ∂-operator on these weighted

L2 spaces of sections of holomorphic bundles over C satisfying the “eigenvalue condi-

tion” above. Roughly speaking, a holomorphic section cannot grow too slowly unless

it is identically zero.

Theorem 3.2.1. Suppose E is a holomorphic vector bundle over C satisfying the

“eigenvalue condition” (3.1) with constants κ0 and ε0. Then, there exists no non-

trivial holomorphic section s ∈ L2(E, e−4ε|z|) for any 0 < ε < ε0/4.


Proof. The proof is a direct cutoff argument. Suppose s is a holomorphic section of

E which belongs to L2(E, e−4ε|z|) for some 0 < ε < ε0/4. We will show that s ≡ 0.

For every real number R > 0, choose a cutoff function φR ∈ C∞c (C) such that

• φR(z) = 1 for |z| ≤ R;

• φR(z) = 0 for |z| ≥ 2R;

• |∇φR| ≤ 2R

.

Let s = φRs. Note that s ∈ C∞c (E). Therefore, by (3.1), properties of φR and

holomorphicity of s,

κ0

∫|z|≤R

|s|2e−4ε|z| dxdy ≤ κ0

∫C|s|2e−4ε|z| dxdy

≤∫C|∂s|2e−4ε|z| dxdy

=

∫C|∂φR|2|s|2e−4ε|z| dxdy

≤ 1

R2

∫R≤|z|≤2R

|s|2e−4ε|z| dxdy

≤ 1

R2‖s‖2

L2(E,e−4ε|z|).

By our assumption, s ∈ L2(E, e−4ε|z|). As R → ∞, the right hand side goes to zero

while the left hand side goes to κ0‖s‖L2(E,e−4ε|z|). Since κ0 > 0, we conclude that

‖s‖L2(E,e−4ε|z|) = 0, hence s ≡ 0. This completes the proof.

3.3 Hormander’s weighted L2 method

In this section, we will use Hormander’s weighted L2 method to construct non-trivial

weighted L2 holomorphic sections on E. Some basic facts on unbounded operators

between Hilbert spaces can be found in [8].

Assume E is a holomorphic vector bundle on C satisfying the “eigenvalue condi-

tion” (3.1) with constants ε0 and κ0. For this section, we also assume that E is the

complexification of some real vector bundle ξ, hence E = E.


Recall that there is a natural first order differential operator ∂ defined on the

space of smooth compactly supported sections of E:

∂ : C∞c (E)→ C∞c (E ⊗ T 0,1C),

where

∂s = (∇ ∂∂zs)⊗ dz.

Let L2(E, e−2ε|z|) be the Hilbert space completion of C∞c (E) with respect to the

weighted L2-norm

‖s‖L2(E,e−2ε|z|) =

(∫C|s|2e−2ε|z| dxdy

) 12

.

Similarly, let L2(E ⊗ T 0,1C, e−2ε|z|) denote the Hilbert space completion of C∞c (E ⊗T 0,1C) with respect to the weighted L2-norm

‖σ‖L2(E⊗T 0,1C,e−2ε|z|) =

(∫C|σ|2e−2ε|z| dxdy

) 12

.

Now, let

∂ : L2(E, e−2ε|z|)→ L2(E ⊗ T 0,1C, e−2ε|z|)

be the maximal closure of ∂ defined as follows: an element s ∈ L2(E, e−2ε|z|) is in the

domain of ∂ if ∂s, defined in the distributional sense, belongs to L2(E⊗T 0,1C, e−2ε|z|).

Then, ∂ defines a linear, closed, densely defined unbounded operator. Note that ∂

is closed because differentiation is a continuous operation in distribution theory. It

is densely defined since Dom(∂) contains all compactly supported smooth sections

C∞c (E), which is clearly dense in L2(E, e−2ε|z|).

By standard Hilbert space theory, the Hilbert space adjoint of ∂, denoted by ∂∗,

is a linear, closed, densely defined unbounded operator and

∂∗

: L2(E ⊗ T 0,1C, e−2ε|z|)→ L2(E, e−2ε|z|).


An element σ belongs to Dom(∂∗) if there is an s ∈ L2(E, e−2ε|z|) such that for every

t ∈Dom(∂), we have

(σ, ∂t)L2(E⊗T 0,1C,e−2ε|z|) = (s, t)L2(E,e−2ε|z|).

We then define ∂∗σ = s.

The key theorem in this section is the surjectivity of ∂.

Theorem 3.3.1. Suppose E is a holomorphic vector bundle over C satisfying the

“eigenvalue condition” (3.1) with constants κ0 and ε0. Moreover, assume that E is

the complexification of some real vector bundle ξ over C. Then,

∂ : L2(E, e−2ε|z|)→ L2(E ⊗ T 0,1C, e−2ε|z|)

is surjective for all 0 < ε < min( ε02,√κ02

).

Proof. By standard Hilbert space theory (see section 4.1 in [8]), it suffices to show

that

(i) the adjoint operator ∂∗

is injective, and

(ii) the range of ∂ is closed.

First of all, we need to compute ∂∗

explicitly.

Claim 1: For any σ = s⊗ dz ∈ C∞c (E ⊗ T 0,1C),

∂∗σ = −∇ ∂

∂zs+ ε

z

|z|s. (3.2)

Proof of Claim 1: This is just integration by parts. Let t ∈Dom(∂), and 〈·, ·〉E,


〈·, ·〉E⊗T 0,1C be the pointwise Hermitian metric on E and E ⊗ T 0,1C respectively.

(σ, ∂t)L2(E⊗T 0,1C,e−2ε|z|) =

∫C〈σ, ∂t〉E⊗T 0,1C e

−2ε|z| dxdy

=

∫C〈e−2ε|z|σ, ∂t〉E⊗T 0,1C dxdy

=

∫C〈e−2ε|z|s,∇ ∂

∂zt〉E dxdy

=

∫C

∂

∂z〈e−2ε|z|s, t〉E dxdy −

∫C〈∇ ∂

∂z(e−2ε|z|s), t〉E dxdy

= −∫C〈∇ ∂

∂zs− ε z

|z|s, t〉E e−2ε|z| dxdy.

= (−∇ ∂∂zs+ ε

z

|z|s, t)L2(E,e−2ε|z|).

The first term in the second to last line vanishes because we have integrated by parts

and used the fact that s is compactly supported. This proves Claim 1.

Next, we need to establish a basic estimate for the adjoint operator ∂∗.

Claim 2: For every 0 < ε < min( ε02,√κ02

), there exists a constant κ1 > 0 such that

κ1

∫C|σ|2e−2ε|z| dxdy ≤

∫C|∂∗σ|2e−2ε|z| dxdy (3.3)

for all σ ∈Dom(∂∗).

Proof of Claim 2: Since E is the complexification of some real vector bundle,

Definition 3.1.1 implies that

κ0

∫C|s|2e−2ε|z| dxdy ≤

∫C|∇ ∂

∂zs|2e−2ε|z| dxdy (3.4)

for any s ∈ C∞c (E). First, we establish claim 2 for σ = s⊗ dz ∈ C∞c (E ⊗ T 0,1C). By


(3.1), the triangle inequality, and that ε <√κ02

, we have

‖∂∗σ‖L2(E,e−2ε|z|) = ‖ − ∇ ∂∂zs+ ε

z

|z|s‖L2(E,e−2ε|z|)

≥ ‖∇ ∂∂zs‖L2(E,e−2ε|z|) − ε‖s‖L2(E,e−2ε|z|)

≥√κ0‖s‖L2(E,e−2ε|z|) − ε‖s‖L2(E,e−2ε|z|)

≥√κ0

2‖s‖L2(E,e−2ε|z|)

=

√κ0

2‖σ‖L2(E⊗T 0,1C,e−2ε|z|).

Squaring both sides give the inequality we want, with κ1 = κ04

. To prove the inequality

for arbitrary σ ∈Dom(∂∗), it suffices to show that C∞c (E⊗T 0,1C) is dense in Dom(∂

∗)

in the graph norm

σ 7→ ‖σ‖L2(E⊗T 0,1C,e−2ε|z|) + ‖∂∗σ‖L2(E,e−2ε|z|).

A proof of this elementary fact can be found in section 3.5. Therefore, we have

completed the proof of claim 2.

Now, claim 2 clearly implies both (i) and (ii) (lemma 4.1.1 in [8]). This finishes

the proof of Theorem 3.3.1.

An important corollary of Theorem 3.3.1 is the following existence theorem of

holomorphic sections of E with controlled growth.

Corollary 3.3.2. Suppose E is a holomorphic vector bundle over C satisfying the

“eigenvalue condition” (3.1) with constants κ0 and ε0. Assume that E is the com-

plexification of some real vector bundle ξ over C.

Then, for any 0 < ε < min( ε02,√κ02

), there exists a non-trivial holomorphic section

s ∈ L2(E, e−4ε|z|), that is,

∂s = 0

and ∫C|s|2e−4ε|z| dxdy <∞.


Proof. First, notice that any holomorphic vector bundle on a non-compact Riemann

surface is holomorphically trivial ([16]). Therefore, we can choose a nowhere vanishing

holomorphic section u of E. However, such a u maybe not be in L2(E, e−4ε|z|). We

will correct it by a cutoff argument and solving an inhomogeneous equation of the

form ∂s = σ to construct a holomorphic section in L2(E, e−4ε|z|).

Take a smooth compactly supported cutoff function ψ ∈ C∞c (C) such that

• ψ(z) = 1 for |z| ≤ 1, and

• ψ(z) = 0 for |z| ≥ 2.

Define

v =1

z

(∂ψ

∂z

)u.

Observe that v ∈ C∞c (E) even though 1/z is singular at z = 0. By Theorem 3.3.1,

there exists w ∈ L2(E, e−2ε|z|) such that

∂w = v ⊗ dz.

Since v is smooth and compactly supported, by elliptic regularity, w is a smooth

section of E (but not necessarily compactly supported).

Next, we let

s = ψu− zw.

Claim: s is a non-trivial holomorphic section in L2(E, e−4ε|z|).

Proof of Claim: First of all,

s(0) = ψ(0)u(0) = u(0) 6= 0

since u is nowhere vanishing. s is holomorphic since

∂s = ∂(ψu)− z∂w =∂ψ

∂zu⊗ dz − z∂w = 0.

Finally, to see that s is in L2(E, e−4ε|z|), we see that ψu is smooth and compactly


supported, hence, in L2(E, e−4ε|z|). Moreover,∫C|z|2|w|2e−4ε|z| dxdy =

∫C|z|2e−2ε|z||w|2e−2ε|z| dxdy

Since |z|2e−2ε|z| ≤ 1 on |z| ≥ R for some R > 0 sufficiently large, it follows that

zw ∈ L2(E, e−4ε|z|), and so does s. This proves our claim and hence establishes the

corollary.

3.4 The main theorem and its proof

In this section, we prove Theorems 3.1.2 and 3.1.3 using the results in section 3.2 and

3.3.

Proof of Theorem 3.1.2: We argue by contradiction. Suppose Theorem 3.1.2 is

false. Then there exists a stable minimal immersion u : Σ → M into an oriented

Riemannian 4-manifold M with uniformly positive isotropic curvature bounded from

below by κ > 0, with Σ uniformly conformally equivalent to C. Recall that u is

minimal if it is a critical point of the area functional with respect to compactly

supported variations, and u is stable if and only if the second variation of area for

any compactly supported variation is nonnegative. Note that we assume u to be an

immersion, therefore we do not allow the existence of branch points.

We consider the normal bundle of the surface u(Σ). We denote by F the bundle

on Σ given by the pullback under u of the normal bundle of u(Σ). Note that F is a

smooth real vector bundle of rank 2. Since M and Σ are orientable, we conclude that

F is orientable. Let FC = F ⊗RC be the complexification of F . Since F is orientable,

the complexified bundle FC splits as a direct sum of two holomorphic line bundles

F 1,0 and F 0,1. Here, F 1,0 consists of all vectors of the form µ(v − iw) ∈ FC, where

µ ∈ C and {v, w} is a positively oriented orthonormal basis of F . An important

observation here is that every section s ∈ C∞(F 1,0) is automatically isotropic, i.e.

(s, s) = 0. Moreover, the splitting F = F 1,0 ⊕ F 0,1 is parallel, i.e. invariant under ∇.

Since u : Σ → M is stable, the complexified stability inequality (see [18]) says


that ∫Σ

⟨R

(s,∂u

∂z

)∂u

∂z, s

⟩dxdy ≤

∫Σ

(|∇⊥∂∂z

s|2 − |∇>∂∂z

s|2) dxdy

for all compactly supported sections s ∈ C∞c (F ), where z = x+iy is a local isothermal

coordinate on Σ. In particular, it holds for all s ∈ C∞c (F 0,1). Note that s ⊥ ∂u∂z

, s

is isotropic and ∂u∂z

is also isotropic (since z is an isothermal coordinate and u is an

isometric immersion). Therefore, {s, ∂u∂z} span a two dimensional isotropic subspace.

Using the lower bound on the isotropic curvature and throwing away the second term

on the right, we get the following inequality

κ

∫Σ

|s|2 da ≤∫

Σ

|∂s|2 da (3.5)

for every s ∈ C∞c (F 1,0), where da is the area element of Σ.

Since Σ is uniformly conformally equivalent to C, by definition, there exists a

diffeomorphism φ : C→ Σ and a constant C > 0 such that

φ∗h = λ2|dz|2 with1

C≤ λ2, (3.6)

where h is the induced metric on Σ. Define E = φ∗(F 1,0) be the pullback of the

holomorphic bundle F 1,0 by φ. Since φ is a conformal diffeomorphism, E is again a

holomorphic bundle over C. By (3.5) and (3.6),

κ

C

∫C|s|2 dxdy ≤

∫C|∂s|2 dxdy (3.7)

for every s ∈ C∞c (E). We then show that E satisfies the “eigenvalue condition” (3.1).

Claim: there exists a constant ε0 > 0 such that for all 0 < ε < ε0,

κ

4C


∫C|∂s|2e−ε|z| dxdy (3.8)

for all compactly supported smooth sections s ∈ C∞c (E).


Proof of Claim: Let s ∈ C∞c (E). Take t = e−ε|z|/2s, notice that

|∂t|2 = |∂(e−ε|z|/2)s+ e−ε|z|/2∂s|2

≤ 2|∂(e−ε|z|/2)s|2 + 2|e−ε|z|/2∂s|2

= 2e−ε|z|(ε2

16|s|2 + |∂s|2)

Applying (3.7) to t ∈ C∞c (E) and using the above estimate,

κ

C

∫C|s|2e−ε|z| dxdy =

κ

C

∫C|t|2 dxdy

≤∫C|∂t|2 dxdy

≤ ε2

8

∫C|s|2e−ε|z| dxdy + 2

∫C|∂s|2e−ε|z| dxdy.

Hence, if we take ε0 = 2√κ√C

, for every 0 < ε < ε0, we get

κ

4C


∫C|∂s|2e−ε|z| dxdy.

This proves our claim.

To summarize, we have constructed a holomorphic vector bundle E over C which

satisfies the “eigenvalue condition” (3.1) with κ0 = κ4C

and ε0 = 2√κ√C

. So we can

apply our results in section 3.2. Moreover, even though E is not a complexification

of a real vector bundle, we see that the result in section 3.3 still holds for E because

E satisfies (3.4) (This follows from the fact that the inequality (3.5) holds with ∂

replaced by ∂, we have to use the fact that F = F 1,0 ⊕ F 0,1 is a parallel splitting).

Hence, if we fix ε > 0 sufficiently small (ε < min( ε04,√κ02

)), then Corollary 3.3.2 gives

a non-trivial holomorphic section s ∈ L2(E, e−4ε|z|), which contradicts Theorem 3.2.1.

This contradiction completes the proof of Theorem 3.1.2.

Proof of Theorem 3.1.3: The proof is very similar to the above. Again we argue

by contradiction. Suppose Theorem 3.1.3 is false. Then there exists a stable minimal


immersion u : Σ→M into a Riemannian n-manifold M with uniformly positive com-

plex sectional curvature bounded from below by κ > 0, with Σ uniformly conformally

equivalent to C.

Let F be the complexified normal bundle of u(Σ) as before and let E = φ∗F . By

our assumption on M , (3.5) holds for every s ∈ C∞c (F ). Exactly the same argument

as above gives our desired contradiction.

3.5 A density lemma

We give a proof of the following density lemma used in the proof of Theorem 3.3.1.

The proof is very similar to that of Lemma 4.1.3 in [29].

Lemma 3.5.1. The subspace C∞c (E⊗T 0,1C) is dense in Dom(∂∗) in the graph norm

σ 7→ ‖σ‖L2(E⊗T 0,1C,e−2ε|z|) + ‖∂∗σ‖L2(E,e−2ε|z|).

Proof. Let σ = s⊗dz ∈Dom(∂∗). First of all, we will show that the set of τ ∈Dom(∂

∗)

with compact support is dense in Dom(∂∗). For each R > 0, let ϕ = ϕR ∈ C∞c (C) be

a smooth cutoff function such that

• ϕ(z) = 1 for |z| ≤ R;

• ϕ(z) = 0 for |z| ≥ 2R;

• |∇ϕ| ≤ 2R

.

Claim 1: When R→∞, ϕR σ converges to σ in the graph norm.

Proof of Claim 1: First of all, we observe that ϕR σ ∈Dom(∂∗). In fact, for any

t ∈Dom(∂),

(ϕRσ, ∂t) = (σ, ∂(ϕRt))− (σ, (∂ϕR)t)

= (∂∗σ, ϕRt)− ((

∂ϕR∂z

)s, t)

= (ϕR∂∗σ − (

∂ϕR∂z

)s, t)


It follows that (ϕRσ, ∂t) is continuous in t for the norm ‖t‖L2(E,e−2ε|z|), so ϕR σ ∈Dom(∂∗)

and

∂∗(ϕRσ) = ϕR∂

∗σ − (

∂ϕR∂z

)s.

Therefore, we have the estimate

|∂∗(ϕRσ)− ϕR∂∗σ| ≤ 2

R|σ|.

Since σ ∈ L2(E ⊗ T 0,1C, e−2ε|z|), and ϕR∂∗σ → ∂

∗σ in L2(E, e−2ε|z|) as R → ∞,

therefore, we have ϕRσ converges to σ in the graph norm as R → ∞. This finishes

the proof of Claim 1.

Next, we need to approximate (in the graph norm) any τ ∈Dom(∂∗) with compact

support by elements in C∞c (E⊗T 1,0C). Take any χ ∈ C∞c (C) with∫C χ dxdy = 1, and

set χδ(z) = δ−2χ(z/δ). Take any τ ∈Dom(∂∗) with compact support, the convolution

τ ∗ χδ is a smooth section of E ⊗ T 1,0C with compact support. Since we can fix a

compact set so that all τ ∗χδ are supported inside the compact set, we see that τ ∗χδconverges to τ in L2(E ⊗ T 1,0C, e−2ε|z|) as δ → 0. It is easy to check that

∂∗(τ ∗ χδ) = (∂

∗τ) ∗ χδ + ε

z

|z|(τ ∗ χδ)− (ε

z

|z|τ) ∗ χδ.

Therefore,

∂∗(τ ∗ χδ)− (∂

∗τ) ∗ χδ = ε

z

|z|(τ ∗ χδ)− (ε

z

|z|τ) ∗ χδ,

and the right hand side converges to 0 in L2(E ⊗ T 1,0C, e−2ε|z|) as δ → 0. Hence, we

conclude that τ ∗ χδ converges to τ in the graph norm as δ → 0. Hence the proof of

Lemma 3.5.1 is completed.

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