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

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.

72pages