Student Geometric & Analysis Seminar (Fall 2023): Mean Curvature Flow

Organizers: Jingbo Wan, Yipeng Wang

Time: Monday 12:30pm--2:00pm

Location:  Room 528 at Columbia Math Department

Extrinsic flows involve the evolution of a submanifold within a larger ambient space, where its velocity is determined by its external curvature. The objective of this seminar is to explore significant extrinsic flows, including the mean curvature flow, Lagrangian mean curvature flow, inverse-mean curvature flow, and more. Mean curvature flow and Ricci flow exhibit numerous analogous characteristics and exert mutual influence. The Lagrangian mean curvature flow was proposed by Thomas-Yau to identify special Lagrangian objects and holds a profound connection with mirror symmetry. Physicists initially introduced the inverse mean curvature flow to demonstrate the monotonicity of Hawking mass, and later, it was utilized by Huisken-Ilmanen to establish the Riemannian Penrose inequality. Throughout the seminar, we will encounter numerous monotonicity formulas applicable to various extrinsic flows, and we will explore their utility in analyzing singularities and proving geometric or physical inequalities. 


[CM12] Colding, Minicozzi Generic mean curvature flow I; generic singularities

[CM15] Colding, Minicozzi Uniqueness of blowups and Lojasiewicz inequalities

[HI] Huisken, Ilmanen The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality 

[N07] Neves Singularities of Lagrangian mean curvature flow: zero-Maslov class case

[N10] Neves Singularities of Lagrangian mean curvature flow: monotone case

[N13] Neves Finite Time Singularities for Lagrangian Mean Curvature Flow 

Title and Abstract (Fall 2023)



Title and abstract

Sep 11th

Yipeng Wang

Stable Shrinker in Mean Curvature Flow 

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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. 

Sep 18th

Yipeng Wang

Classification of F-Stable Shrinkers 

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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.  

Sep 25th

Jingbo Wan

Rigidity of contracting map using harmonic map flow 

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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.  

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Dec 4th