Abstract

The problem of the regularity of minimal surfaces was one of the main driving forces behind the development of geometric measure theory and was, via De Giorgi, partly responsible for the development of elliptic regularity theory. The main part of this talk will discuss the regularity theory of minimal submanifolds via the notion of stationary, integral varifolds. These objects form an important class of very weakly defined, ‘low-regularity’ minimal surfaces and were introduced in their modern form by Allard in the early 70’s. Going via a description of Allard’s Regularity Theorem, we will build towards a discussion of some new work-in-progress.

This talk is intended to be accessible and will begin with the definition of a Minimal Submanifold of Euclidean space as a stationary point of the area functional.