Regularity theory of minimal surfaces in Euclidean space

Prerequisite

Real analysis, some knowledge of elliptic PDE would be helpful but not required.

Abstract

The goal of this course is to present the proof of the following remarkable result in the regularity theory of codimension one minimal surfaces in the Euclidean space: the singular set of a locally area minimizing hypersurface in n dimensional Euclidean space has zero (n-1) dimensional Hausdorff measure. The proof presented in the course is due to De Giorgi. We shall cover the theory of the Caccioppoli sets and prove the key De Giorgi lemma for the minimal Caccioppoli set. If the time permits, we shall proceed to show that the dimension of the singular set cannot exceed n-8. The lectures will mainly follow the reference "Minimal Surfaces and Functions of Bounded Variation" by Enrico Giusti.

Lecturer Intro.

Dr. Pengyu Le graduated from ETH Zürich in 2018, then became a Van Loo postdoctoral fellow in University of Michigan. He joined BIMSA as an assistant research fellow in 2021. His research interest lies in differential geometry and general relativity.