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(Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents

Max Engelstein, Luca Spolaor and Bozhidar Velichkov

Geometry & Topology 23 (2019) 513–540
Abstract

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing any given trace in the radial direction along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (eg work of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (eg integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new regularity result for almost area-minimizing currents at singular points where at least one blowup is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of Leon Simon (1983), but independent from it since almost-minimizers do not satisfy any equation.

Keywords
regularity of minimal surfaces, epiperimetric inequality, almost area-minimizing currents
Mathematical Subject Classification 2010
Primary: 53A10
References
Publication
Received: 3 May 2018
Accepted: 30 August 2018
Published: 5 March 2019
Proposed: Tobias H Colding
Seconded: Bruce Kleiner, Gang Tian
Authors
Max Engelstein
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Luca Spolaor
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Bozhidar Velichkov
Laboratoire Jean Kuntzmann
Université Grenoble Alpes
Grenoble
France