Second order rectifiability of integral varifolds of locally bounded first variation

TL;DR

This paper proves that integral varifolds with locally bounded first variation can be decomposed into countably many C^2 submanifolds, with mean curvature matching almost everywhere, advancing understanding of their geometric regularity.

Contribution

It establishes second order rectifiability of integral varifolds with bounded first variation, showing they can be covered by smooth submanifolds with consistent mean curvature.

Findings

Integral varifolds with bounded first variation are covered by countably many C^2 submanifolds.

Mean curvature of submanifolds matches that of the varifold almost everywhere.

Advances the regularity theory of varifolds in geometric measure theory.

Abstract

In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywhere with respect to the varifold.