题目：Singularities and generic regularity of minimal surfaces

报告人：李阳垟 (芝加哥大学)

时间：2025.06.10 (周二) 15:30-17:10

    2025.06.11 (周三) 15:30-17:10

    2025.06.12 (周四) 9:00-10:40

    2025.06.13 (周五) 9:00-10:40

地点：理科楼A304

摘要：

Singularities are a common and often inevitable feature across many areas of mathematics, including algebraic geometry, differential topology, geometric measure theory, partial differential equations, and dynamical systems. In the study of minimal surfaces, singularities naturally arise when the ambient manifold has dimension at least eight. This phenomenon was first revealed in a seminal 1968 paper by J. Simons, where the singularity is modeled on the now-famous Simons’ cone.

A central question in the regularity theory of minimal surfaces is how these singularities behave under perturbations of the ambient metric—specifically, whether they can be eliminated by a generic perturbation. An early affirmative answer was given by R. Hardt and L. Simon in 1985 (see also the application by N. Smale in 1993), who prove the existence of smooth minimal foliation near certain area-minimizing cones, such as Simon's cone. Yet, this result does not conclude the story: the classification of all area-minimizing cones remains open, and many singular cones are not area-minimizing.

In these lectures, I will discuss the theory of generic regularity for minimal surfaces and its recent developments, focusing in particular on my recent joint work with Zhihan Wang (Inventiones mathematicae, 2025), in which we prove the generic regularity of all minimal hypersurfaces in 8-dimensional closed manifolds. Time permitting, I will also present an application of the generic regularity to the non-persistence of Hsiang's minimal spheres, based on joint work with Alessandro Carlotto and Zhihan Wang.

报告人简介：李阳垟，2013-2017年就读于清华大学数学科学系并获学士学位，2016年获清华大学本科生特等奖学金；2017-2022年在普林斯顿大学数学系攻读博士学位，师从国际著名几何学家Fernando Codá Marques教授；2022年至今担任芝加哥大学数学系L.E. Dickson讲师。研究方向为微分几何与偏微分方程，相关论文发表在Inventiones Mathematicae, Journal of Differential Geometry, Geometry & Topology等国际著名数学期刊。

邀请人：马辉