Rectifiability of the free boundary for varifolds

TL;DR

This paper proves partial rectifiability of the free boundary of varifolds by refining variation measures and relating mean curvature integrability to the structure of the boundary, advancing geometric measure theory.

Contribution

It refines the understanding of first variation measures for varifolds with free boundary and links mean curvature integrability to boundary rectifiability.

Findings

First variation of varifold with free boundary is a Radon measure.

Set where density does not exist or is infinite has Hausdorff dimension at most k-p.

Part of the first variation with positive, finite density is (k-1)-rectifiable.

Abstract

We establish a partial rectifiability result for the free boundary of a $k$-varifold $V$. Namely, we first refine a theorem of Gr\"uter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature $H$ of $V$ is in $L^p$ for some $p \in [1,k]$, then the set of points where the $k$-density of $V$ does not exist or is infinite has Hausdorff dimension at most $k-p$. We use this result to prove, under suitable assumptions, that the part of the first variation of $V$ with positive and finite $(k-1)$-density is $(k-1)$-rectifiable.