Analysis Seminar

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The Allard regularity problem

Speaker: Camillo De Lellis, Institute for Advanced Study

Location: Warren Weaver Hall 1302

Date: Thursday, March 20, 2025, 11 a.m.

Synopsis:

Stationary integral varifolds, introduced by Almgren in the sixties, are a very useful generalization of minimal surfaces, which play an important role in a variety of geometric problems. While all known examples of nonsmooth stationary integral varifolds consist of pieces of classical minimal surfaces coming together at a codimension set of singularities, the only general regularity result available is the 1972 celebrated regularity theorem of Allard, which shows that the regular part of the varifold is dense in its support. Even proving that the singular part of 2-dimensional ones in R3 has zero 2-dimensional measure is surprisingly challenging. In this talk, I will explain what the difficulties are, propose some conjectures which we hope might simplify the problems, and present some partial results towards their solution, which anyway deliver some interesting structural consequences. The talk is based on two joint works with Camillo Brena, Stefano Decio, and Federico Franceschini.