Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities

Abstract

In this paper, we use optimal mass transportation to derive an isoperimetric inequality and an interior Michael-Simon-Sobolev inequality for weakly twice differentiable integral varifolds of dimension and codimension at least $2$, with locally bounded first variation and $L^2_{loc}$ mean curvature. This generalizes a result of Brendle and Eichmair. In both inequalities the corresponding constants are optimal in codimension 2. As part of this process we also generalize Alexandrov's theorem for convex functions to this setting. This necessitates the use of measure and distribution-valued solutions to certain geometric differential operators, in addition to the theory of $L^p$ Taylor expansions started by Calderón and Zygmund.