Organizers: Jingbo Wan, Yipeng Wang
Time: Monday 12:30pm2:00pm
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, inversemean 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 ThomasYau 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 HuiskenIlmanen 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.
References:
[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: zeroMaslov class case
[N10] Neves Singularities of Lagrangian mean curvature flow: monotone case
[N13] Neves Finite Time Singularities for Lagrangian Mean Curvature Flow
Date 
Speaker 
Title and abstract 

Sep 11th 
Yipeng Wang 
Stable Shrinker in Mean Curvature Flow Show/hide AbstractsA shrinker is the time slice of a selfsimilar 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 Ffunctional. In this talk, we will briefly discuss this approach and prove that closed Fstable shrinkers are mean convex. 
Sep 18th 
Yipeng Wang 
Classification of FStable Shrinkers Show/hide AbstractsWe will finish the proof of Fstable shrinkers are mean convex from last time. Then we will discuss certain curvature estimates for noncompact shrinkers and show that Fstable shrinkers can only be sphere or hyperplane. 
Sep 25th 
Jingbo Wan 
Rigidity of contracting map using harmonic map flow Show/hide AbstractsIn this talk, we are going to consider the rigidity of map between positively curved closed manifolds, which is motivated by the recent work of TsaiTsuiWang. We show that distance nonincreasing 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 nonincreasing and area nonincreasing. This is based on the harmonic map heat flow and it partially answer a question raised by TsaiTsuiWang. This is a joint work with Prof. ManChun Lee in CUHK. 
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