Minimal surfaces and mean curvature flow

Lecture series organized by Hojoo Lee (autumn@kias.re.kr)

When? 15(Thr) - 16 & 27(Tue) - 28 May Where? Room 1424, KIAS

Topics

1. Mean curvature flow of hypersurfaces (6 talks)

– Robert Haslhofer (Courant Institute of Mathematical Sciences, NYU)

2. Minimal surfaces, shrinkers and translators (4-� talks)

– Hojoo Lee (Korea Institute for Advanced Study)

3. Maximum principle for minimal hypersurfaces with singularities

– Leobardo Rosales (Korea Institute for Advanced Study)

2 Robert Haslhofer · Hojoo Lee · Leobardo Rosales

1. Mean curvature flow of hypersurfaces

Robert Haslhofer (Courant Institute of Mathematical Sciences, NYU)

II. Abstract. A family of hypersurfaces evolves by mean curvature flow ifthe velocity at each point is given by the mean curvature vector. In the last15 years, White developed a far-reaching structure theory for weak solutionswith positive mean curvature, and Huisken-Sinestrari constructed a flow withsurgery for hypersurfaces where the sum of the smallest two principal curvaturesis positive. In this course, I will present joint work with Kleiner, where we give astreamlined and unified treatment of the theory of White and Huisken-Sinestrari.After reviewing the necessary background, I’ll discuss our new a priori estimatesfor mean convex mean curvature flow and mean curvature flow with surgery.They are based on the beautiful noncollapsing result of Andrews, which says thatthe condition of admitting interior and exterior balls of radius α/H is preservedunder the flow. In the last two lectures, I’ll sketch how we use our estimates toobtain the main structural results for mean convex level set flow, and to prove theexistence of mean curvature flow with surgery.

References[1] B. Andrews, Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16 (2012),

1413–1418.

[2] R. Haslhofer, B. Kleiner, Mean curvature flow of mean convex hypersurfaces, 2013, http://arxiv.org/abs/1304.0926, (Lecture Video – http://goo.gl/8caYbo).

[3] R. Haslhofer, B. Kleiner, Mean curvature flow with surgery, 2014, http://arxiv.org/abs/1404.2332, (Lecture Video – http://goo.gl/eN5Laf)

[4] G. Huisken, C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces,Invent. Math. 175 (2009), 137–221.

[5] B. White, The size of the singular set in mean curvature flow of mean convex sets, J. Amer.Math. Soc. 13 (2000), 665–695.

[6] B. White, The nature of singularities in mean curvature flow of mean convex sets. J. Amer.Math. Soc. 16 (2003), 123–138.

http://arxiv.org/abs/1304.0926http://arxiv.org/abs/1304.0926http://goo.gl/8caYbohttp://arxiv.org/abs/1404.2332http://arxiv.org/abs/1404.2332http://goo.gl/eN5Laf

Minimal surfaces and mean curvature flow in Euclidean space 3

2. Minimal varieties in higher codimension

III. Minimal surfaces, shrinkers and translators in Euclidean space

Hojoo Lee (Korea Institute for Advanced Study)

This is the third in a series of my lectures on submanifolds with zero meancurvature vector field in Riemannian manifolds. As described in [1–3], the theoryof minimal submanifolds (in higher dimension and codimension) is breathtakinglybeautiful, and admits lots of unsolved problems.

We begin with N > 10 equivalent defintions of minimal surfaces, and willfocus on translaing solitions to mean curvature flow in Euclidean space. Wesay that a surface in R3 is the translator when its mean curvature vector fieldagrees with the normal component of a constant Killing vector field in R3.Translators arise as Hamiltons convex eternal solutions and Huisken-Sinestrari’sType II singularities for the mean curvature flow. We explicitly will sketch theconstruction of various translators, and concretly describe that translators arenovel generalization of minimal surfaces and constant mean curvature surfaces.

References[1] Meeks, William H., III; Pérez, Joaquı́n, The classical theory of minimal surfaces, Bull.

Amer. Math. Soc. (N.S.) 48 (2011), no. 3, 325–407.

[2] Schoen, Richard, Minimal submanifolds in higher codimension, XIV School on Dif-ferential Geometry, Mat. Contemp. 30 (2006), 169–199.

[3] Yau, Shing-Tung, Perspectives on geometric analysis, Geometry and analysis. No. 2,417–520, Adv. Lect. Math. (ALM), 18, Int. Press, Somerville, MA, 2011.

4 Robert Haslhofer · Hojoo Lee · Leobardo Rosales

3. Maximum principle for minimal hypersurfaces withsingularities

Leobardo Rosales (Korea Institute for Advanced Study)

The classical maximum principle implies the difference between two solutionsto the minimal surface equation over an open region which agree at an inte-rior point cannot attain an interior maximum or minimum, unless if those twosolutions are equal. This can be restated to conclude that if two minimal hy-persurfaces coincide at a point near which one hypersurface lies on one sideof the other, then those two hypersurfaces coincide. To what extent is this trueif we allow our hypersurfaces to have singularities? We present here the workof Wickramasekera, who sharply answers this question based on his importantresults on the regularity of stable minimal hypersurfaces.

References[1] T. Ilmanen, A strong maximum principle for singular minimal hypersurfaces, Calc. Var.

Partial Differential Equations, 4 (1996), no. 5, 443–467.

[2] N. Wickramasekera, A sharp strong maximum principle and a sharp unique continuationtheorem for singular minimal hypersurfaces, Calc. Var. Partial Differential Equations (2013):1–14.

[3] N. Wickramasekera, A general regularity theory for stable codimension 1 integral varifolds,A general regularity theory for stable codimension 1 integral varifolds, Ann. of Math.(2) 179 (2014), no. 3, 843–1007.

Lecture Series on Geometry and Analysishttp://ultrametric.weebly.com/monet-seminar.html

http://ultrametric.weebly.com/monet-seminar.html

Mean curvature flow of hypersurfacesMinimal varieties in higher codimensionMaximum principle for minimal hypersurfaces with singularities