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. 

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: 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)