Fully Nonlinear PDEs and RelatedGeometric Problems

Bo Guan

Ohio State University

Xiamen University

Institute of Mathematics, AMSS

Beijing

1

2

The three well known classical PDEs are Laplace equation,heat equation, and wave equation, representing the three typesof PDEs with rather distinct properties: elliptic, parabolic andhyperbolic equations, respectively.

• Laplace equation – elliptic

∆u :=∂2u

∂x21

+ · · ·+ ∂2u

∂x2n

= 0

• Heat equation – parabolic

ut = ∆u

• Wave equation – hyperbolic

utt = ∆u

3

In general, an elliptic PDE of the second order can be writtenin the form

(1) F (∇2u,∇u, u, x) = 0.

The ellipticity means

(2) F ij[u] ≡ F ij(∇2u,∇u, u, x) > 0

where, if we write F (A, ·, ·, ·) and A = aij ∈ Sn×n,

F ij ≡ ∂F

∂aij

Sn×n is the set of n by n symmetric matrices. Equivalently, thelinearized operator

Lu = F ij[u]∇i∇j + lower order terms

is elliptic.

4

Equation (2) is

• linear, if F is linear in u, ∇u and ∇2u;otherwise, nonlinear;• semilinear, if F is linear in ∇u and ∇2u;• quasilinear, if F is linear in ∇2u;• fully nonlinear, if F is not linear in ∇2u.

5

Let’s first look at some examples of nonlinear equations fromgeometry.

• The minimal surface equation

div( Du√

1 + |Du|2)

= 0.

For n = 2,

(1 + u2y)uxx − 2uxuyuxy + (1 + u2

x)uyy = 0.

• The minimal surface equation in hyperbolic space(δij −

uiuj1 + |Du|2

)uij =

n

u.

• The spacelike maximal surface equation in Minkowski space

div( Du√

1− |Du|2)

= 0.

The spacelike condition

|Du| < 1.

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• The Monge-Ampere equation

detD2u = ψ.

For n = 2,uxxuyy − u2

xy = ψ.

• The complex Monge-Ampere equation

det∂2u

∂zi∂zj= ψ.

• The prescribed Gauss curvature equation

detD2u = K(1 + |Du|2)n+22 .

For spacelike hypersurfaces in Minkowski space

detD2u = K(1− |Du|2)n+22 .

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There are important fourth order equations in geometry.

• Willmore surface equation.• Affine maximal hypersurface equation

uij∂2

∂xi∂xjdetD2u = 0.

8

1. The Isoperimetric Inequality. Let C be a simple closedcurve in R2. Then

4πA ≤ L2

where A is the enclosed area, L donotes the length of C.More generally, let Ω be a domain in Rn. Then

ω1(n|Ω|)n−1 ≤ |∂Ω|n

where ω donotes the volume of the unit sphere in Rn.

Proof. Consider the Neumann problem

∆u = C in Ω,∂u

∂ν= 1 on ∂Ω.

Let Γ+ be the lower contact of (the graph of) u

Γ+ = x ∈ Ω : u(y) ≥ u(x) +Du(x) · (y − x).

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It is easy to see that

B1 ⊂ Du(Γ+)

where B1 denote the unit ball in Rn centered at the origin. Con-sequently,

|B1| ≤ |Du(Γ+)| ≤∫

Γ+

detD2u ≤ 1

nn

∫Γ+

(∆u)n ≤ 1

nnCn|Ω|.

By the Divergence Theorem,

C|Ω| =∫

Ω

∆u =

∫∂Ω

∂u

∂ν= |∂Ω|.

So

C =|∂Ω||Ω|

.

This completes the proof.

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2. Alexandrov Theorem. An embedded closed hypersurfacein Rn+1 of constant mean curvature must a sphere.

The proof: Alexandrov’s moving plane method, based the max-imum priciple. It is also called Alexandrov reflection principle.The idea can be explained using curves of contant curvature inthe plane.

Let C be a simple closed curve in R2. Locally,

C = graph of f.

The curvature of C is

κ =f ′′

(1 + |f ′|2) 32

.

This is an ODE for f . Suppose κ is constant. Then C musta circle, following the uniqueness of solution to the initial valueproblem.

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3. Minkowski Type Problems. Let Σ be a hypersurface inRn+1. We use κ = (κ1, . . . , κn) to denote the principal curvaturesof Σ with respect to its interior normal. The kth Weingartencurvature Wk of Σ is defined as

Wk = σk(κ1, . . . , κn), k = 1, . . . , n

where σk is the kth elementary symmetric function. For k = 1, 2and n, Wk corresponds to the mean, scalar and Gauss curvature,respectively.

Suppose now that Σ is a strictly convex closed hypersurface.The Gauss map n : Σ → Sn is then a diffeomorphism. Let n−1

denote its inverse which we call the inverse Gauss map.

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In 1950s, A. D. Alexandrov and S. S. Chern raised the followingextended version of the Minkowski problem

Let 1 ≤ k ≤ n be a fixed integer, and ψ > 0 on Sn. Does thereexist a closed strictly convex hypersurface Σ in Rn+1 such that

(3) Wk(n−1(x)) = ψ(x) ∀ x ∈ Sn?

For k = n this is the classical Minkowski problem, which wasstudied by Minkowski, Alexandrov, Lewy, Nirenberg, Pogorelov,Cheng-Yau, etc.

Theorem 1. For k = n, a necessary and sufficient condition is

(4)

∫Sn

x

ψ(x)= 0.

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This turns out not to be the case for 1 ≤ k < n.

Theorem 2 (P.-F. Guan, G. 2002). (a) For every 1 ≤ k < nand any nonzero real number m, there exists a parameter familyof closed strictly convex hypersurfaces (all are small perturbationsof the unit sphere) in Rn+1 satisfying

(5)

∫Sn

x

(Wk(n−1(x)))m6= 0.

(b) There exists a function f ∈ C∞(Sn) and a constant δ > 0such that for all t ∈ (0, δ), problem (3) has no solution for ψ :=(1 + tf)−1 while (4) is satisfied.

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A partial existence result.

Theorem 3. Suppose ψ is invariant under an automorphic groupG of Sn without fixed points, i.e., ψ(g(x)) = ψ(x) for all g ∈ Gand x ∈ Sn. Then there exists a closed strictly convex hypersurfaceΣ in Rn+1 satisfying (3).

For instance, if ψ(−x) = ψ(x) for all x ∈ Sn the problem issolvable.

The PDE:σn(λ)

σn−k(λ)=

1

ψ

where λ = λ(∇2u+ ug) = eigenvalues of ∇2u+ ug on Sn.Subsequent work: Sheng-Trudinger-Wang.

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4. Plateau Type Problems. Let f be a smooth symmetricfunction of n (n ≥ 2) variables, and

Γ = Γ1, . . . ,Γma disjoint collection of closed smooth embedded submanifolds ofdimension (n− 1) in Rn+1.

Question. Does there exist an immersed hypersurfaces M inRn+1 of constant curvature

(6) f(κ[M ]) = K

with boundary

(7) ∂M = Γ?

Here κ[M ] = (κ1, . . . , κn) denotes the principal curvatures of Mand K is constant.

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The Plateau problem: f = σ1, the mean curvature of M , raisedby Joseph-Louis Lagrange in 1760, named after Joseph Plateauwho experimented with soap films, and solved independently byJesse Douglas and Tibor Rado in 1930’s. But there were a lotof subsequent developments and research activities, especially ingeometric measure theory.

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For f = σn, the Gauss curvature, M is locally determined by

detD2u = K(1 + |Du|2)n+22 .

This equation is elliptic for strictly convex solutions. We requireM to be locally strictly convex, i.e, the second fundamental formis M is positive definite everywhere.

• The second fundamental form of each Γk is nondegenerateeverywhere. For n = 1 this means that the curvature of Γknever vanishes.• There are topological obstructions (H. Rosenberg).

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Existence results.

• Caffarelli-Nirenberg-Spruck (1980’s): The Dirichlet prob-lem is solvable over a strictly conevx domain, provided thatthere is a strictly convex subsolution.• Spruck-G. (1993, 1998): On any smooth bouned domain,

as long as there is a strictly convex subsolution.• Spruck-G. (2002): If Γ bounds a locally strictly convex

hypersurface, it bounds one with constant Gauss curvature.It was also independently proved by Trudinger-Wang (2002).

• Spruck-G. (2004): This is true for more general function f .

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Some of the technical issues.

• The Dirichlet problem in general domain.• Perron’s mehtod for locally convex hypersurfaces.• Local gradient estimates.• Area minimizing for locally convex hypersurfaces.

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Some general questions to understand.

• Global bahavior/properties of solutions—Liouville type theorem—Bernstein theorem—Symmetry

• Existence/expression of solutions.—Separation of variables, eigenfunction expansions—Poisson representation—d’Alembert’s formula—A priori estimates

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In the rest of this talk we shall mainly concerned with equationsof the form

(8) f(λ(A[u])) = ψ

on a Riemannian manifold (Mn, g), where

• f : a smooth symmetric function of n variables defined inΓ ⊂ Rn

• Γ: a symmetric open and convex cone with vertex at theorigin, ∂Γ 6= ∅, and Γn ⊆ Γ where

(9) Γn ≡ λ ∈ Rn : each component λi > 0. ⊆ Γ.

• A[u] = ∇2u+ χ.• λ(A) = (λ1, · · · , λn) denotes the eigenvalues of A.

Note that F (A) = f(λ(A)) if and only if F (PAP T ) = F (A) forany orthogonal matrix P .

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Examples of f . This covers a very broad class of equations.

• f = σ1k

k or (σk/σl)1k−l define on

Γk = λ ∈ Rn : σj(λ) > 0, 1 ≤ j ≤ k

where σk(λ) is the elementary symmetric function

σk(λ) =∑

i1<...<ik

λi1 . . . λik, k = 1, . . . , n.

In particular, σ1(λ) = ∆u, σn(λ) = det∇2u.• The special Lagrange equation: f(λ) =

∑tan−1 λi.

• f = logPk where

Pk(λ) :=∏

i1<···<ik

(λi1 + · · ·+ λik), 1 ≤ k ≤ n

defined in the cone

Pk := λ ∈ Rn : λi1 + · · ·+ λik > 0.

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• The inverse sum:

(10) f(λ) = −∑ 1

λαi, λ ∈ Γn, α > 0.

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The fundamental structure conditions [CNS1985]. Tostudy the equations under the framework of elliptic PDE theory,we need some basic assumptions.

• ellipticity

(11) fi = fλi ≡∂f

∂λi> 0 in Γ, 1 ≤ i ≤ n,

• concavity

(12) f is a concave function in Γ

• nondegeneracy:

(13) infΩψ > sup

∂Γf

wheresup∂Γ

f ≡ supλ0∈∂Γ

lim supλ→λ0

f(λ).

These conditions were introduced by Caffarelli-Nirenberg-Spruckin 1985 and have become standard in the literature.

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Admissible functions. A function u ∈ C2 is called admissibleif λ(A[u]) ∈ Γ.

• (11): Eq (8) is elliptic for admissible solutions.• (12): F (A) ≡ f(λ[A]) is concave for A with λ[A] ∈ Γ.• (13): Eq (8) will not become degenerate.• (11) & (13) & |∇2u| ≤ C: Eq (8) becomes uniformly ellip-

tic.• Evans-Krylov theorem: Suppose that (1) is uniformly

elliptic, F is concave w.r.t. ∇2u and |u|C2(Ω) ≤ C. Then

|u|C2,α(Ω) ≤ C.

• Schauder theory: C2,α estimates imply higher regularity.• Continuity method: |u|C2,α(Ω) ≤ C implies the classical

solvability of the Dirichelt problem.

From this point of view, conditions (11)-(13) are fundamental tothe classical solvability of equation (8).

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The ultimate goal is to solve equation (8). To prove the exis-tence of classical solutions.

• The Dirichlet problem.• On closed manifolds.

The key is to derive global C2 estimates. We hope to establishthis for general manifolds—without curvature restrictions, andfor general domians in the case of the Dirichlet problem—withoutassumptions on the geometric shape of ∂M , the boundary of M .

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Question: Are assumptions (11)-(13) necessary? Sufficient?

• For the degenerate Monge-Ampere equation, the solutionmay fail to belong to C1,1(Ω).• The Dirichlet problem for detD2u = 1 in Ω ⊂ Rn withu = 0 on ∂Ω does not have a solution unless Ω is strictlyconvex.• Nadirashvili et al.: For nonconcave F , the solution may

fail to belong to C1,1 (n ≥ 5).• CNS3: There is an equation(n = 2) satisfying (11)-(13),

with solution in C∞(B1) ∩ C(B1) but not in C1(B1).

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Previous Work.Caffarelli, Nirenberg and Spruck (1985)Chou-Wang (2006)IvochkinaIvochkina-Trudinger-Wang (degenerate case)Kryov (1980’s)Y.-Y. Li (1990)Trudinger (1996)Urbas (2003)

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The Dirichlet problem in Rn.

• Caffarelli, Nirenberg and Spruck (1985, CNS).

Theorem 4 ([CNS3, Acta 1985]). Let Ω be a bounded smoothdomain in Rn, n ≥ 2, ψ ∈ Ck,α(Ω), ϕ ∈ Ck+2,α(∂Ω), k ≥ 2.Assume a) (11)-(13), and in addition b), c), d) below. Then theDirichelt problem

(14)

f(λ(∇2u+ χ) = ψ in Ω

u = ϕ on ∂Ω

admits a unique admissible solution u ∈ Ck+2,α(Ω)Moreover, if ψ ∈ C∞(Ω), ϕ ∈ C∞(∂Ω) then u ∈ C∞(Ω).

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The additional conditions:

b) for every C > 0 and λ ∈ Γ there is a number R = R(C, λ)such that

(15) f(λ1, . . . , λn−1, λn +R) ≥ C

(16) f(Rλ) ≥ C

c) A geometric condition on ∂Ω:

(17) (κ1, . . . , κn−1, R) ∈ Γ on ∂Ω for some R > 0

where (κ1, . . . , κn−1) are the principal curvatures of ∂Ωd) χ = 0.

• For the Monge-Ampere equation (f = σ1/nn ), c) means that

Ω is strictly convex.

Trudinger in 1995 removed condition (15).

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• Guan (2014, for general domains).

Theorem 5 (Guan 2014, arXiv:1403.2133). Assume

a) (11)-(13) hold,e) the subsolution assumption: there exists an admissible

subsolution u ∈ C2(Ω)

(18)

f(λ(∇2u+ χ) ≥ ψ in Ω

u = ϕ on ∂Ω.

The Dirichlet problem (14) then has a unique admissible solutionu ∈ Ck+2,α(Ω).

Moreover, if ψ ∈ C∞(Ω), ϕ ∈ C∞(∂Ω) then u ∈ C∞(Ω).

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The proof. To derive

|u|C2(Ω) ≤ C.

Then C2,α and higher order estimates follows from Evans-KrylovTheorem and Schauder theory; existence by the continuity method.

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Theorem 6 (Guan, 2014). Let u ∈ C4(M) ∩ C2(M) be an ad-missible solution of the Dirichlet problem (14). Suppose (11)-(13)hold and that there exists an admissible subsolution u ∈ C2(M):

(19)

f(λ[∇2u+ χ]) ≥ ψ in M,

u = ϕ on ∂M.

Then

(20) maxM|∇2u| ≤ C.

The proof consists of two steps:

• a maximum principle for |∇2u|

(21) maxM|∇2u| ≤ C

(1 + max

M|∇u|2 + max

∂M|∇2u|

)• the boundary estimate

(22) max∂M|∇2u| ≤ C.

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The concavity and subsoltuion. For σ > sup∂Γ f , let

Γσ = λ ∈ Γ : f(λ) > σand suppose Γσ 6= ∅. By (11) and (12) the level set

∂Γσ = λ ∈ Γ : f(λ) = σi.e., boundary of Γσ, is a smooth convex hypersurface.

For λ ∈ ∂Γσ let

νλ =Df(λ)

|Df(λ)|be the unit normal to ∂Γσ and Tλ∂Γσ denote the tangent planeof ∂Γσ at λ.

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Theorem 7. Let µ ∈ Γ and β > 0. There exists uniform constantε > 0 such that when |νµ − νλ| ≥ β,

(23)∑

fi(λ)(µi − λi) ≥ f(µ)− f(λ) + ε∑

fi(λ) + ε.

• Used in global estimates for |∇2u| too.• Apply Theorem 7 to µ = λ[u], λ = λ[u].

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Equations on closed Riemannian manifolds. Let (Mn, g)be a Riemannian manifold without boundary.

Y.-Y. Li (1990) first studied equation (14) for χ = g on closedRiemannian manifolds with nonnegative sectional curvature. JohnUrbas (2002) removed the curvature assumption.

Their main extra assumptions are

lim|λ|→∞

∑fi =∞

andlim|λ|→∞

∑fi(1 + λ2

i ) =∞

respectively.

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• The notaion of subsolutions on closed Riemannianmanifolds. When M is closed, the subsolution assumption doesnot make sense. Indeed, a subsolution must be a solution, or thereis no solution. This is a consequence of the maximum principle.

The tangent cone at infinity (Guan, DJM 2014). Define

Sσµ = λ ∈ ∂Γσ : νλ · (µ− λ) ≤ 0and

C+σ = µ ∈ Rn : Sσµ is compact.

which we call the tangent cone at infinity to Γσ.

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Theorem 8 (Guan, DJM 2014). a) C+σ is open.

b) Assume (11)-(13) and that there exists u ∈ C2(M) with

(24) λ(∇2u+ χ)(x) ∈ C+ψ(x), ∀ x ∈ M.

Then

(25) maxM|∇2u| ≤ C

(1 + max

∂M|∇2u|

)

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The enlarged cone C+σ . We now construct a larger cone from

C+σ . Note that the unit normal vector of any supporting hyperplan

to Γσ belongs to Γn. We define C+σ to be the region in Rn bounded

by those supporting hyperplans to C+σ with unit normal vector in

∂Γn; so C+σ = Rn if there are no such supporting planes. Clearly, if

C+σ 6= Rn it is an open symmetric convex cone with vertix at a1 for

some a ∈ Rn where 1 = (1, . . . , 1) ∈ Rn. Moreover, µ + Γn ⊂ C+σ

for µ ∈ C+σ and C+

σ ⊂ C+ρ if σ ≥ ρ.

Theorem 9 (Guan, 2015). Let µ ∈ C+σ and d(µ) denote the

distance from µ to ∂C+σ . Then there exist δ, ε > 0 such that for

any λ ∈ ∂Γσ, either

(26) fi(λ) ≥ δ∑

fk(λ)

or

(27) fi(λ)(µi − λi) ≥ ε∑

fi(λ) + ε.

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Theorem 10 (Guan, 2015). Assume (11)-(13) and that thereexists u ∈ C2(M) with

(28) λ(∇2u+ χ)(x) ∈ C+ψ(x), ∀ x ∈ M.

Then

(29) maxM|∇2u| ≤ C

(1 + max

∂M|∇2u|

).

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Szekelyhidi (2015) introduced the another nontion of general-ized subsolutions:

(30) (λ(∇2u+ χ)(x) + Γn) ∩ ∂Γψ(x) is compact ∀ x ∈ M.

and proved the same estimates under a little stronger assump-tions. The following result clarify the ralations.

Theorem 11 (Guan, 2016). a) For Type I cone, C+σ = C+

σ . b)Assumptions (28) and s (30) are equivalent.

According to CNS, a cone Γ is Type 1 if each λi-axis belongsto ∂Γ. For instance, Γk (k ≥ 2) are Type 1.

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• Canonical χ: χ = 0 or χ = g. For instance,

(31) det(∇2u+ g) = K(x)(−2u− |∇u|2) det g

is the Darboux equation (isometric embedding).• ∇2u+Ricg: the Bakry-Emery Ricci tensor of the Riemann-

ian measure space (Mn, g, e−udg).• Te Ricci soliton equation: ∇2u+ Ricg = λg• ∇2u+ug on Sn. (In classical geometry. Minkowski problem,

extensions proposed by Alexandrov, Chern; Christoffel-Minkowskiproblem.)• In conformal geometry. The Schouten tensor of (Mn, e2ug)

(32) χ = du⊗ du− 1

2|∇u|2g + Sg

where Sg is the Schouten tensor of (Mn, g); the Ricci tensor

A[u] = ∇2u+ γ∆ug + du⊗ du− 1

2|∇u|2g + Ricg.

• In optimal transportation.• Conformal deformation of metrics on Hermitian manifolds.

A[u] := ±Ricu =√−1∂∂u± Ric.

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More general equations. For the more general equation

F (D2u+ χ) = ψ,

we assume the function F to be defined in an open convex cone Γin Sn×n, the (inner product) space of n× n symmetric matrices,with vertex at 0, Γ+ ⊆ Γ 6= Sn×n where Γ+ denotes the coneof positive matrices, and to satisfy the fundamental structureconditions:

(a) the ellipticity condition.(b) the concavity condition.

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Equations on complex manifolds.

Let (Mn, ω) be a compact Hermitian manifold of complex di-mension n ≥ 2 with smooth boundary ∂M which may be empty(M is closed) and let, for a function u ∈ C2(M), χ[u] := χ(·, du(·), u(·))be a real (1, 1) form on M , and define

χu := χ[u] +√−1∂∂u.

The equation:

(33) f(λ(χu)) = ψ(z, du, u) in M

where λ(χu) = (λ1, · · · , λn) denote the eigenvalues of χu withrespect to the metric ω.

This covers most of the important equations in complex geom-etry.

• In local coordinates:

χu =√−1(uij + χij)dzi ∧ dzj.

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• Guan-Nie: χu := χ[u] +√−1∂∂u where χ depends lin-

early on du.• Guan-Qiu-Yuan: A[u] = ∆uω − γ

√−1∂∂u + χ where

χ = χ(du).

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• The complex Monge-Ampere equation:

(34) χnu = ψnωn

corresponds to equation (8) for f = σ1/nn . It plays a central role

in Kahler geometry.

• Calabi-Yau Theorem. Yau’s proof of Calabi conjecture;Aubin indepedently for c1(M) < 0.• Extension to Hermitian case.

– Cherrier: n = 2;– Tosatti-Weinkove.

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• The Dirichlet problem.– Caffarelli-Kohn-Nirenberg-Spruck: strongly pseudocon-

vex domain in Rn;– Guan (1998): general domain in Cn;– Guan adn Qun Li (2010): on Hermitian manifolds.

• Some applications.– Chern-Levine-Nirenberg conjecture: Pengfei Guan– Donaldson’s conjecture on geodesics in the space of

Kahler metrics: Mabuchi, Donaldson, Xiuxiong Chen, manyothers .....

– Pluricomplex Green functions.– Totally real submanifolds.

48

There have also been increasing interests in other fully nonlinearequations from Kahler geometry.• Donaldson:

(35) χnu = ψχn−1u ∧ ω, χu > 0,

proposed in the setting of moment maps, where he assumes χ isalso Kahler and ψ is the Kahler class invariant

ψ = c1 =

∫M χn∫

M χn− ∧ ω.

This corresponds to f = σn/σn−1. The equation is also closelyrelated to the lower bound and properness of the Mobuchi energy.

49

• Conformal metrics on Hermitian manifolds. Let α =e±uω be a conformal metric on M . The Chern-Ricci form is givenby

±Ricα = ±√−1∂∂ logαn =

√−1∂∂u± Ricω

Consequently, the problem of determining a metric in the con-formal class of ω with special properties of the Chern-Ricci formleads to the following equation for χ = ±Ricω

(36) F( χue±uω

)= ψ on M.

The sign in front of Ricα is determined by requiring χu ∈ Γ. Thenegative sign case will be much more difficult to study.

For F (A) = (detA)1/n, this equation is related to a conjectureof Yau on the holomorphic sectional curvature of a Kahler mani-folds which was recently solved by Wu-Yau and Tosatti-Yang.

50

• Balanced metrics with prescribed volume form Re-call that a Hermitian metric ω is balanced if d(ωn−1) = 0 andGauduchon if ∂∂(ωn−1) = 0. A well know result due to Gaudu-chon asserts that any conformal class of Hermitian metrics on acompact (closed) complex manifold contains a Gauduchon met-ric. Fu-Wang-Wu introduced an equation of prescribed volumefor balanced metrics, which can be described as below followingTosatti-Weinkove.

Let ω0 be a balanced metric on a closed Hermitian manifold(M,ω). We seek a balanced metric η such that

ηn−1 = ωn−10 +

√−1∂∂(uωn−2)

for some u ∈ C∞(M) and prescribed volume

(37)ηn

ωn= ψ on M.

This is a Monge-Ampere type equation of (n− 1, n− 1) forms.Nevertheless, by the Hodge star duality approach of Tosatti-Weinkove, equation (37) can be converted to an equation of form

51

(38) for F (A) = Pn−1(A) where χ depends linearly on u and thegradient of u; More precisely,

χ = (trχ− (n− 1)χ

where

χ =1

(n− 1)!?(ωn−1

0 −√−1∂u∂(ωn−2)+

√−1∂u∂(ωn−2)+

√−1u∂∂(ωn−2))

and ? denotes the Hodge star-operator.

52

• Gauduchon conjecture.Let complex dimension n ≥ 2 and let Ω be a closed real (1, 1)

form on M with [Ω] = cBC1 (M) in the Bott-Chern cohomologygroup H1,1

BC(M, R). In 1984, Gauduchon conjectured that thereexists a Gauduchon metric ω on M with Chern-Ricc curvatureRicω = Ω.

This is a natural extension of the Calabi conjecture for Kahlermanifolds solved by Yau. It was discovered by Tosatti and Weinkoveand independently by Popovicithat the Gauduchon conjecture re-duces to solving a Monge-Ampere type equation of the form

(38) det(Φu) = eF+b det(ωn−1) in M

with

Φu = ωn−10 +

√−1∂∂u ∧ ωn−2 + Re(

√−1∂u ∧ ∂ωn−2) > 0

and supM u = 0, where ω0 is any Hermitian metric (positive def-inite (1, 1) form) and ω a Gauduchon metric. This equation isequivalent to one for f = Pn−1 with linear dependences of χ ondu.

53

Results of Guan-Xiaolan Nie.We study equation (38) on closed Hermitain manifold with χ

depending linearly on du. Our result applies to Pn−1 giving theestimates need in proving Gaudochun’s conjecture.

Theorem 12. Let ψ ∈ C2(M) and u ∈ C4(M) be an admissiblesolution of (8). Suppose that there exists a function u ∈ C2(M)satisfying

(39) λ(χu(z)) ∈ C+ψ(z) ∀ z ∈M,

and that at any fixed point on M where gij = δij and gij = δijλiwith λ1 ≥ · · · ≥ λn,

(40)∑

fi(|χi1,ζα|+ χii,ζα1|) ≤ Cλ1fα, ∀α ≤ n− r0

where r0 = min

rank of C+ψ(z) : z ∈M

. Then

(41) maxM|∆u| ≤ C1e

C2(u−infM u)

where C1 depends on |∇u|C0(M) and C2 is a uniform constant(independent of u).

54

Results of Guan-Qiu-Yuan.

F (∆uω + γ√−1∂∂u+ χ(z, u, du)) = ψ, γ < 1.

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Thank You!