A shrinker is the time slice of a self-similar shrinking solution of the Mean Curvature Flow, which is crucial for analyzing singularity models. It turns out that a shrinker can also be treated as a solution to a certain variational problem called the F-functional. In this talk, we will briefly discuss this approach and prove that closed F-stable shrinkers are mean convex. 

We will finish the proof of F-stable shrinkers are mean convex from last time. Then we will discuss certain curvature estimates for non-compact shrinkers and show that F-stable shrinkers can only be sphere or hyperplane.  

In this talk, we are going to consider the rigidity of map between positively curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially answer a question raised by Tsai-Tsui-Wang. This is a joint work with Prof. Man-Chun Lee in CUHK.  

Shrinkers can be represented as minimal hypersurfaces in Euclidean space equipped with a conformal metric. As a result, classical techniques for studying minimal surfaces are also applicable to the analysis of mean curvature flow. In this presentation, we will review some classical a priori estimates of the second fundamental form on minimal hypersurfaces and discuss a Schoen-Simon-Yau-type inequality for mean convex shrinkers. 

In this talk, we'll discuss a paper by Brendle-Hung-Wang that applies a monotonicity formula of inverse mean curvature flow to prove a Minkowski inequality in AdS-Schwarzschild manifolds. Despite the lack of full convergence in this context (IMCF on Asymptotically hyperbolic manifold), we establish a roundedness estimate that helps estimate the lower bound of a monotone quantity along the flow. Besides IMCF, Professor Brendle's geometric inequality and a sharp Sobolev inequality on the standard sphere are key components of the proof. 

We continue the discussion about IMCF and a Minkowksi inequality in AdS-Schwarzschild manifold. Last time, we went over the basic a priori estimates for IMCF using both the parametric and non-parametric versions of the flow. In particular, short time existence and long time existence were clarified. In this talk, we will delve into an improved roundness estimate for the flow and the monotonicity formula along the flow. It's worth noting that IMCF doesn't fully converge in asymptotically hyperbolic manifolds. To overcome this, we will rely on Beckner's sharp Sobolev inequality on standard spheres and a geometric inequality by Professor Brendle to estimate the lower bound of the monotone quantity as we approach the limit. 

Varifolds provide measure-theoretic generalizations of smooth manifolds that arise naturally in the study of minimal surfaces and mean curvature flow due to their compactness properties. In this talk, we introduce the main concepts of stationary varifolds, the measure-theoretic analogue of minimal surfaces, and their generalized mean curvature. We hence state Allard’s regularity theorem; this asserts that near a point where the growth of balls is sufficiently close to Euclidean and the generalized mean curvature is locally Lp, the varifold is locally the graph of a C1,α function with estimates. We present a recent proof due to de Philippis-Gasparetto-Schulze, following ideas of Savin on viscosity solutions of the minimal surface system. 

Varifolds provide measure-theoretic generalizations of smooth manifolds that arise naturally in the study of minimal surfaces and mean curvature flow due to their compactness properties. In this talk, we introduce the main concepts of stationary varifolds, the measure-theoretic analogue of minimal surfaces, and their generalized mean curvature. We hence state Allard’s regularity theorem; this asserts that near a point where the growth of balls is sufficiently close to Euclidean and the generalized mean curvature is locally Lp, the varifold is locally the graph of a C1,α function with estimates. We present a recent proof due to de Philippis-Gasparetto-Schulze, following ideas of Savin on viscosity solutions of the minimal surface system. 

A soliton is a special solution to a partial differential equation that maintains its shape and moves at a constant velocity. In the context of mean curvature flow, a translating soliton is a solution to the mean curvature flow equation that moves by a constant velocity in the direction of a vector. Translating solitons are particularly interesting because they provide insights into the behavior of evolving surfaces. On the other hand, we say that a surface is semi-graphical if when we remove a discrete set of vertical lines, then the resulting surface is the graph of a smooth function. We are going to provide a classification of all the semi-graphical translator in Euclidean 3-space. First, we will describe a comprehensive zoo of all examples of this type of translators, and then we will focus on classification arguments. We will conclude with some open problems. This talk summarizes various joint works with D. Hoffman and B. White, on one hand, and with M. Saez and R. Tsiamis, on the other. 

A soliton is a special solution to a partial differential equation that maintains its shape and moves at a constant velocity. In the context of mean curvature flow, a translating soliton is a solution to the mean curvature flow equation that moves by a constant velocity in the direction of a vector. Translating solitons are particularly interesting because they provide insights into the behavior of evolving surfaces. On the other hand, we say that a surface is semi-graphical if when we remove a discrete set of vertical lines, then the resulting surface is the graph of a smooth function. We are going to provide a classification of all the semi-graphical translator in Euclidean 3-space. First, we will describe a comprehensive zoo of all examples of this type of translators, and then we will focus on classification arguments. We will conclude with some open problems. This talk summarizes various joint works with D. Hoffman and B. White, on one hand, and with M. Saez and R. Tsiamis, on the other. 

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case, which was first proved by Huisken-Ilmanen using the Inverse mean curvature flow in dimension three. As the first part of this series of talks, we are going to discuss some basic notions and the Geroch monotonicity of Hawking mass under smooth inverse mean curvature flow. After this brief introduction, we will focus on the Huisken-Ilmanen’s weak formalism of inverse mean curvature flow, which is the essential ingredient in the proof of Riemannian Penrose inequality. 

Last time, we discussed the Geroch monotonicity of Hawking mass under smooth inverse mean curvature flow that can lead to Riemannian Penrose Inequality. We also gave a brief overview of Huisken-Ilmanen’s argument. In this talk, we aim to delve deeper into the intricacies of Huisken-Ilmanen’s weak formalism concerning the inverse mean curvature flow. To provide a more comprehensive understanding, we will focus on the freezing and variation approach, essential components in defining a weak solution for the level set inverse mean curvature flow. This approach not only grants us analytical advantages but also plays a crucial role in demonstrating the existence of weak solutions. To enhance our geometric understanding of weak IMCF, we will also cover an equivalence formulation utilizing sublevel sets. With both the analytic and geometric understanding of the weak IMCF, we will briefly talk about the heuristic reason for Geroch monotonicity along weak IMCF.  

This talk explores an alternative method for establishing the Riemannian Penrose inequality through Bray’s conformal flow. The conformal flow preserves the non-negativity of scalar curvature and the outermost minimizing property of the inner boundary. By using the positive mass theorem, one can demonstrate that the ADM mass is non-increasing under Bray’s conformal flow, eventually converging to half of Schwarzschild space. This convergence implies the desired Riemannian Penrose inequality. Notably, Bray’s approach offers advantages over IMCF approach, as it can be generalized to dimensions less than eight due to the absence of Gauss-Bonnet and enhances the inequality by replacing the area of the largest component of the horizon with the total area of the horizon, owing to fewer topology restrictions on the flow.