Bi-Lipschitz Regularity of 2-Varifolds with the Critical Allard Condition

TL;DR

This paper proves that 2-varifolds satisfying the critical Allard condition are locally bi-Lipschitz equivalent to flat disks, under small area deviation and mean curvature conditions.

Contribution

It establishes bi-Lipschitz regularity for 2-varifolds under the critical Allard condition, extending regularity results to this specific setting.

Findings

Varifolds are bi-Lipschitz homeomorphic to flat disks.

Small area deviation implies regularity.

Mean curvature control leads to regularity.

Abstract

For an intergral $2$-varifold $V=\underline{v}(\Sigma,\theta_{\ge 1})$ in the unit ball $B_1$ passing through the original point, assuming the critical Allard condition holds, that is, the area $\mu_V(B_1)$ is close to the area of a unit disk and the generalized mean curvature has sufficient small $L^2$ norm, we prove $\Sigma$ is bi-Lipschitz homeomorphic to a flat disk in $\mathbb{R}^2$ locally.