Seminars

Title :   Noncompactness of the constant scalar curvature and constant boundary mean curvature equation

Abstract : Suppose $(M,g_0)$ is a compact Riemannian manifold with boundary $\partial M$. As a generalization of the Yamabe problem, one can consider the problem of finding a conformal metric to $g_0$ with constant scalar curvature in $M$ and constant boundary mean curvature on $\partial M$. In this talk, I will consider some noncompactness result of this problem. If time permits, I will mention some existence results obtained by geometric flow. 

Title :   Min-max constructions of prescribed topological type on 3-manifolds

Abstract : While Almgren-Pitts Min-max theory have provided the development of finding infinitely minimal hypersurfaces on closed manifolds, Simon-Smith variant of min-max theory has contributed to obtain the minimal surfaces with topological control on 3-manifolds. We discuss the existence and regularity theory of minimal surfaces with topological control. In particular, we discuss equivariant min-max theory and the recent existence theorem of Wang-Zhou on four minimal 2-spheres on 3-spheres with generic metric. If time permits, I will also cover the relation with other min-max theory, and topological applications, and related topics.

Title :   Free boundary and capillary embedded geodesics on Riemannian 2-disks with a strictly convex boundary 

Abstract : The existence of embedded geodesics on surfaces is the foundational problem. I will explain the existence of two free boundary embedded geodesics on Riemannian 2-disks with a strictly convex boundary by free boundary curve shortening flow on surfaces, which is a free boundary analog of Grayson’s theorem in 1989. I will then explain the existence of two capillary embedded geodesics on Riemannian 2-disks with a strictly convex boundary with a certain condition via multi-parameter min-max construction. Finally, I will explain the Morse Index bound of such geodesics.

Title :   Constructing special Lagrangian submanifolds by gluing (Dec. 1)

Gluing construction has been successful in constructing solutions to geometric PDEs, such as  minimal surfaces, CMC surfaces, Ricci-flat metrics, etc, by desingularizing simple but singular ‘standard pieces’. I will explain the general philosophy behind gluing constructions and the applications in constructing special Lagrangian submanifolds.

Title :   Gluing constructions in Lagrangian mean curvature flow: translating solitons and infinite-time singularities (Dec.4)

We will continue the discussion in gluing constructions. This time focusing on constructing solutions to LMCF, including my recent results on new examples of  Lagrangian translators, and solutions with infinite-time singularities.

Title:   Introduction to Lagrangian mean curvature flow

Abstract:  Lagrangian mean curvature flow (LMCF) is a canonical way of deforming Lagrangian submanifolds in Calabi-Yau manifolds, with the goal of constructing special Lagrangian submanifolds - an important  class of volume-minimizing submanifolds. In this talk, I will review the basic concepts  and provide an overview of the recent developments in Lagrangian mean curvature flow and related topics.  

Title:   Geometric Inequalities: A Survey and Open Problems  

Abstract:  We will discuss some old and new open problems in geometric analysis, related to isoperimetric inequalities. 

Title:  On the singular and long-time behavior of inverse mean curvature flow

Abstract:   In these lectures, I will discuss some of my work on the inverse mean curvature flow (IMCF) of closed hypersurfaces in Euclidean space. IMCF is an extrinsic curvature flow that expands mean-convex hypersurfaces by mean curvature and arises naturally in geometric analysis. We will focus on two questions: (1) Which types of initial data develop finite-time singularities under IMCF, and how does the solution behave near singularities? and (2) Which types of initial data do not develop finite-time singularities under IMCF, and how do those solutions behave at large times?

For question (1), I will show that there exist finite-time singularities of inverse mean curvature flow in which the total curvature of the evolving hypersurface remains uniformly bounded up to the singular time. This behavior contrasts sharply with the singular behavior of other curvature flows such as mean curvature flow. For question (2), I will show that certain families of non-star-shaped initial data exist for all time and homothetically converge to spheres under IMCF. Both of these results are based in part on a more general criteria for singularity formation and asymptotic convergence for inverse mean curvature flow.

Talk 1.

Title: Geometric applications of Wirtinger-type inequalities

Abstract: We prove various sharp geometric inequalities for closed curves on the Euclidean plane. In particular, we obtain both sharp lower and upper bounds for the isoperimetric deficit.

Talk 2.

Title: Analytic applications of Fourier analysis

Abstract: Using Fourier analysis, we prove a high order generalization of Wirtinger-type inequalities.

Title:  Stable minimal hypersurface in R^4 by Otis-Li

Abstract:  I will provide an outline of the method presented in Otis-Li paper (https://arxiv.org/abs/2108.11462) that proves the classical conjecture stating that a complete, two-sided, stable minimal hypersurface in R^4 is flat.  I will be focusing more on the Stern Bochner method for harmonic level set used in the proof of that paper. 

Reference: Otis’s lecture note (Sec 8.5 - 8.8) https://web.stanford.edu/~ochodosh/Math258-min-surf.pdf

Title:   Gallery of minimal surfaces (극소 곡면 그림 미술관)  

Abstract: 미국 수학자 더글라스에게 첫 번째 필즈 메달을 안겨 준 물리학자 플라토의 비누막 실험 문제부터, 푸앵카레의 위상수학 숙제를 푼 업적으로 준다는 필즈 메달을 받으러 오지도 않았던 러시아 수학자 페렐만의 기하학적 미분방정식론에 이르기까지, 오랜 세월 동안 사랑받아 온 극소곡면 이론이 생물학, 물리학, 화학, 재료공학, 건축학등 다양한 분야들을 연결하는 아름다움에 취해 보자. 특히, 이 강의에서는 주기를 갖는 극소곡면들의 여러가지 예들과 성질들을 살펴본다.  

Title:  Scalar curvature rigidity of polyhedron in hyperbolic 3-space and generalizations 

Abstract: Gromov proposed in 2014 the dihedral rigidity conjecture to define a notion of scalar curvature with a lower bound in the weak sense. Chao Li proved some cases of the dihedral rigidity conjecture in the Euclidean case and the cube case via free boundary minimal surface and capillary surface.

In this talk, I will generalize the argument to the hyperbolic 3-space via constant mean curvature surfaces with capillary boundary conditions. I also cover some recent work in progress of scalar curvature rigidity of convex sets in both the Euclidean and the hyperbolic case with certain symmetry. The talk is based on joint works with Gaoming Wang (Cornell). 

Title: Yamabe problem and Yamabe flow I, II

Abstract: In the first lecture, I will explain what the Yamabe problem is.  I will talk about how one can solve the Yamabe problem from PDE point of view. In the second lecture,  I will talk about the Yamabe flow, which is the geometric flow introduced to study the Yamabe problem.  Then I will mention some basic properties of the Yamabe flow.  In particular,  I will talk about the Yamabe soliton, which is the self-similar solution of the Yamabe flow.  If time permits,  I will also talk about the Yamabe problem with boundary. 

Title: Higher order asymptotics and convergence rate of minimal hypersurfaces. 

Abstract:  A minimal hypersurface is often asymptotic to a minimal cone at near a singularity or its spatial infinity.  Similarly, solutions to elliptic and parabolic equations exhibit self-similar behaviors near their spatial and temporal limits. In this series of talks, we will investigate a method, based on the theory of spectral analysis and dynamical system, to derive higher order asymptotics and a rate of convergence toward the limit self-similar solution. The talks will mostly be presented in the setting of minimal hypersurface, but the method generally applies to diverse static and time-evolution equations and it was crucial tool in recent progresses in classification problems. 

Title: Some rigidity results for compact hypersurfaces with planar boundaries in Hyperbolic space. 

Abtract: We prove the Heintze-Karcher type inequality for capillary hypersurfaces supported on a totally geodesic hyper-plane in hyperbolic space, and equality case only occurs on capillary totally umbilical hypersurfaces. Then we apply this result to prove the Alexandrov type theorem for embedded capillary hypersurfaces in $\mathbb H_+^{n+1}$. Besides, we will give some other rigidity results for immersed capillary hypersurfaces.

Title: An introduction to ancient solutions to the mean curvature flow.  

Abstract: The mean curvature flow (MCF) is a family of immersions moved by the mean curvature vector. A solution to MCF is called ancient if it is defined for all negative time. In this series of talks, we give a brief introduction to MCF and discuss some recent research on the study of the ancient solutions to the MCF. This includes (i) the classification of embedded convex solutions in one dimension, (ii) a half space property for ancient solutions, and (iii) some explicit gluing constructions of one-dimensional examples.  

Title : Analysis of positive solutions to one-dimensional generalized double phase problems 

Abstract : In this talk, we study positive solutions to the one-dimensional generalized double phase problems. First, we show various existence results including the existence of at least two or three positive solutions according to the behaviors of nonlinear term near zero and infinity. Both positone and semipositone problems are considered, and the results are obtained through the Krasnoselskii type fixed point theorem. We also apply these results to show the existence of positive radial solutions for high-dimensional generalized double phase problems on the exterior of a ball. Second, we deal with more general problems which are not suitable $C^1[0,1]$ space and require more delicate analysis.

Title: An abstract critical point theorem with applications to elliptic problems with combined nonlinearities

Abstract:

We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical p-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional p-Laplacian problems.

Mirror Symmetry Correspondence of Punctured Spheres and Degenerate Cusps

We give a new mirror of punctured Riemann spheres as degenerate cusp singularities, which are commutative rings. Then we illustrate the explicit correspondence between geodesics in the (hyperbolic) punctured sphere and Cohen-Macalay modules over its mirror ring, based on Burban-Drozd's classification (2017). This is a joint work with Cheol-Hyun Cho, Wonbo Jeong and Kyoungmo Kim.

Title: Inverse mean curvature flow with a free boundary in hyperbolic space (2)

In the second talk, I am going to present proofs of the convergence result in the geodesic ball of hyperbolic space and formulate some problems which interest me. 

• The talk will be given in English.

Title: Inverse mean curvature flow with a free boundary in hyperbolic space (1)

I am going to talk about inverse mean curvature flow with a free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.  I am going to review basics of mean curvature type flows and monotone quantities under inverse mean curvature flow. 

• The talk will be given in English.

Title: Myers-type theorems for Bakry-Emery Ricci tensor

One of the natural and important topics in Riemannian geometry is the relation between curvature and topology. Examples of good explanations of the relationship between curvature and topology include comparison geometry,  Cheeger-Gromoll splitting theorems,  Myers theorems,  etc. In particular,  Myers theorems show that positive Ricci curvature has strong topological consequences. In this talk, we focus on the Myers theorems. We review the classical Myers theorem and related theorems for the Ricci curvature and we also look at the results in the Bakry-Emery Ricci tensor.

Strichartz estimate is known as one of the fundamental tools to study the nonlinear dispersive PDEs and has been fruitfully used to prove the well-posedness of their Cauchy problem. In this talk, we review the known results of the classical Stricharz estimates for the Schrodinger equation. We then discuss the weighted Strichartz estimates introduced in our recent work with a spatial power weight |x|^{−α}, α>0. Interestingly, in an application of these weighted estimates, the well-posedness theory for the inhomogeneous nonlinear  Schrodinger equation (INLS) was established in the critical case which was left unsolved until recently.

In this talk, I will talk about what the weighted Yamabe problem is and mentioned some related results that Jinwoo Shin (KIAS) and I obtained.

Timetable :  

08. Feb. Tuesday

Mario Chan

Talk 1 : What is so special about Kahler manifolds?

Jaehoon Lee

Talk 1 : Introduction to Spin representation

Deahwan Kim

Maximal graphs via the shearing construction

Keomkyo Seo

Partially overdetermined boundary value problems in a convex cone

Sunghong Min

Fundamental tone of complete weakly stable cmc hypersurfaces

09. Feb. Wednesday

Mario Chan 

Talk 2 : What is so special about Kahler manifolds?

Jihyeon Lee

Algebraic backgrounds on spin geometry

Yuan Shyong Ooi

Characterization of self-shrinker with low index

Sungho Park

Capillary surfaces in a triangular cone

Mikyoung Lee

Regularity theory for second order elliptic equations

10. Feb. Tuesday

Mario Chan

Talk 3 : What is so special about Kahler manifolds?

Eungbeom Yeon

Talk 3 : Dirac operator and proof of the positive energy theorem

Sangwoo Park

Complete self-shrinkers in 3-dimensional Euclidean space

Juncheol Pyo

Evolution of the first eigenvalue along the inverse mean curvature flow in space forms

Title and Abstract

Riemannian manifolds with positive scalar curvature (Jaehoon Lee)

In this talk, we discuss some elementary properties of scalar curvature and introduce specific examples with positive scalar curvature metrics.

Sewing 3-manifolds with positive scalar curvature (Jaehoon Lee)

Basilio, Dodziuk, and Sormani introduced a new technique called sewing based on the tunnel construction of Gromov-Lawson and Schoen-Yau. In this time, we discuss the technique in detail and talk about its role in convergence problems. The talk will follow some parts of the paper:

J. Basilio, J. Dodziuk, and C. Sormani, Sewing Riemannian manifolds with positive scalar curvature. J. Geom. Anal., 28 (2018), 3553-3602.

Gromov-Lawson surgery method on manifolds with positive scalar curvature (Eungbeom Yeon)

We look into Gromov Lawson surgery method on manifolds with positive scalar curvature.

Compact manifolds with negative scalar curvature (Eungbeom Yeon)

We show that every compact manifold M of dimension greater or equal to 3 admit a metric with negative scalar curvature.

Rigidity of area minimizing tori in $3$-manifolds of nonnegative scalar curvature (Jihyeon Lee)

In 1980, Fischer, Colbrie and Schoen conjectured that if $(M,g)$ is a complete $3$-manifold with nonnegative scalar curvature and if $\Sigma$ is a two-sided torus in $M$ which is suitably of least area, then $M$ is flat. After 20 years, in 2000, Cai and Galloway presented a proof of this conjecture assuming $\Sigma$ is of least area in its isotopy class. In this talk, we will prove Cai-Galloway's proof for this conjecture, and to prove this we will follow Otis Chodosh's lecture note.

Curvature estimates and Bernstein type theorem of f-minimal hypersurfaces in Minkowski space (Daehwan Kim)

The prescribed mean curvature submanifolds have great physical importance both in the Riemannian geometry and in the pseudo-Riemannian geometry. For example, space-like hypersurfaces with a prescribed mean curvature were constructed as the stationary limits of a geometric evolution equation. In the first talk, we show a particular inequality related to the mean curvature and several curvature estimates of a space-like hypersurface in the Minkowski space. In the second talk, there is a well-known result of Cheng and Yau: the only complete maximal hypersurface is a hyperplane in the Minkowski space. A extended result for the f-minimal space-like hypersurface with a restricted condition of the weight function f is provided.

Bochner formula and its application to harmonic function on manifold with non-negative Ricci curvature (Ooi, Yuan Shyong)

In this lecture, I will discuss the classical result of Bochner formula and how we can use it to study harmonic function on manifold with non-negative Ricci curvature. In particular we discuss how to apply Bochner formula to obtain gradient estimate and Harnack inequality for harmonic function. 

Stern-Bochner formula with geometric application on three manifold (Ooi, Yuan Shyong)

This lecture aims to study a more recent Bochner technique due to Daniel L. Stern. We will apply this Stern-Bochner technique to study the connection between level set of harmonic function and the scalar curvature of three manifold. 

Non parabolicity of stable minimal hypersurfaces in Euclidean space (Sangwoo Park)

We first talk about  non parabolicity of general complete non compact manifolds. Then we use the parabolicity to characterize stable minimal hypersurfaces in Euclidean space.

The Green’s function on a stable minimal hypersurface (Sangwoo Park)

We establish several useful facts about the behavior of the level sets of such a Green’s function. 

Two classic results about ends of minimal hypersurfaces in the Euclidean space (Eungmo Nam)

In this talk, we study two classic results about ends of minimal hypersurfaces in the Euclidean space. More specifically, we first deal with the result of Cao-Shen-Zhu, which states that a complete oriented stable minimal hypersurface immersed in the Euclidean space must have only one end. And then we review the result of Li-Wang which asserts that  a complete oriented minimal hypersurface immersed in the Euclidean space with finite index has finitely many ends. Note that we consider these theorems in terms of parabolicity of ends of minimal hypersurfaces.

Consider a variational problem for a geometric functional such as the area or the volume.  Generalizing the derivative test in elementary calculus, one computes the first and second variations. Consequently, the Morse index intuitively gives the number of distinct deformations in which decrease the functional to the second order. If there is a constraint, how does the index change? In this talk, we’ll answer that question in a general abstract framework and then apply it to study capillary surfaces. This is joint work with Detang Zhou.

Self-shrinking plane curves were completely determined by Abresch and Langer, and each curve satisfies a transcendental relation. In this talk, we will review how the curves can be determined by the transcendental relation. We then discuss the classification problem for closed Lagrangian self-shrinkers in R^4. In particular, we will show that closed Lagrangian self-shrinkers symmetric to a hyperplane satisfy similar transcendental relations and can be completely determined. We will also talk about a possible approach without symmetry assumptions.

In this talk, we concern rigidity results for free boundary zero mean curvature hypersurfaces in a geodesic ball and analogous results for self-shrinkers.

First, we show that any minimal hypersurface with free boundary in a closed geodesic ball in a round open hemisphere S^{n+1} which is Killing-graphical is a geodesic disk. We note that we do not assume any topological condition on the hypersurface. Second, we provide an analogous result of free boundary maximal hypersurfaces in a region bounded by a de Sitter space in the Lorentz–Minkowski space. More precisely, any smooth, compact free boundary maximal hypersurface in a de Sitter ball is the spacelike coordinate planar disk passing through the center of the de Sitter space. Notably, there is no more condition such as the killing-graphical. Therefore free boundary maximal hypersurfaces in the de Sitter space are substantially rigid.

Third, we consider  free boundary self-shrinkers in an Euclidean ball. We show that any graphical self-shrinker with free boundary in a ball centered at the origin in R^{n+1} is a flat disk passing through the origin. It is an analogous result for self-shrinkers of the mean curvature flow.

Studying the Gauss map of a minimal surface is crucial to understanding minimal surfaces in Euclidean spaces. Classical results and recent results will be reviewed on the subject. Main material will be theorem of H.Fujimoto which states that the Gauss map of a complete nonflat minimal surface in R^3 can omit at most four points of the sphere. Generalization of the result to R^n by Fujimoto-Osserman-Ru will be also considered. Later on we will talk about recent developments on the subject by A.Alarcón, I. Fernández and F. López.

In this talk, we are going to consider regularity theory of elliptic and parabolic equations on the manifold. We will try to understand the influence of geometry and challenges. We also try to discuss the relation between degeneracy of PDE, geometry, and distortion of probability density function.

In this course, we develop a few mathematical approaches to the Fraser-Li conjecture and review some of the related results. Talks will be divided into three parts as follows.

Talk 1. Applying holomorphic methods to the Fraser-Li conjecture 

Talk 2. Applying PDE-theoretic methods to the Fraser-Li conjecture

Talk 3. Capillary minimal surfaces outside the unit ball

The motion of hypersurfaces under the mean or Gaussian curvature is a geometric evolution that has been intensively investigated in the last few decades. The first talk introduces basic notions and describes how the hypersurface evolves by the curvature flows. In particular, we discuss developments of singularities. In the second talk, we introduce some free boundary problems in curvature flows. Among them, curvature flows with obstacles that exist for a long time without singularities will be mainly presented. This is based on a joint work with Ki-Ahm Lee.

The motion of hypersurfaces under the mean or Gaussian curvature is a geometric evolution that has been intensively investigated in the last few decades. The first talk introduces basic notions and describes how the hypersurface evolves by the curvature flows. In particular, we discuss developments of singularities. In the second talk, we introduce some free boundary problems in curvature flows. Among them, curvature flows with obstacles that exist for a long time without singularities will be mainly presented. This is based on a joint work with Ki-Ahm Lee.

Caffarelli-Hardt-Simon constructed embedded minimal hypersurface in Euclidean space which has an isolated singularity but which is not a cone in 1984. The example constructed by them is also asymptotic to a given arbitrary non-planar minimal cone. In this 3 series of talks, I will discuss the higher codimension analog of their construction. 

I will divide the talk into three parts. In the first part, I will discuss about the classical result in hypersurface case regarding the Dirichlet problem of minimal surface equation. Then I will talk about the pioneer work of Lawson and Osserman in the higher codimension setting. For the second part I will discuss a more recent work by Mu-Tao Wang, Yng- Ing Lee and Mao-Pei Tsui about the stability problem. For the third talk, I will talk about my recent joint- work with Yng-Ing Lee and Mao-Pei Tsui on the uniqueness problem of minimal surface system.