# Second order rectifiability of varifolds of bounded mean curvature

## TL;DR

This paper proves that varifolds with bounded mean curvature and density bounds are almost everywhere covered by countably many smooth submanifolds, extending geometric and PDE techniques to varifold regularity.

## Contribution

It introduces a novel approach combining stochastic geometry and viscosity solutions to establish second order rectifiability of varifolds.

## Key findings

- Varifolds with bounded mean curvature are covered by smooth submanifolds almost everywhere.
- The method extends curvature notions to arbitrary closed sets.
- The approach bridges stochastic geometry and PDE theory in geometric measure theory.

## Abstract

We prove that the support of an $ m $ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $ \mathscr{H}^{m} $ almost everywhere by a countable union of $m$ dimensional submanifolds of class $ \mathcal{C}^{2} $. We obtain this result using the notion of curvature of arbitrary closed sets originally developed in stochastic geometry and extending to our geometric setting techniques developed by Trudinger in the theory of viscosity solutions of PDE's.