Global regularity of integral 2-varifolds with square integrable mean curvature

TL;DR

This paper establishes sharp criteria for integral 2-varifolds to be represented by smooth conformal immersions with controlled mean curvature, linking varifold regularity to Willmore energy minimization.

Contribution

It provides new regularity criteria for 2-varifolds with critical mean curvature integrability, connecting varifold theory with smooth surface immersions and Willmore energy minimization.

Findings

Varifolds with energy below a threshold are curvature varifolds with bounded second fundamental form.

The approach links varifold regularity to classical surface minimization problems.

Smooth solutions to the Willmore problem are obtained via varifold minimization.

Abstract

We provide sharp sufficient criteria for an integral $2$-varifold to be induced by a $W^{2,2}$-conformal immersion of a smooth surface. Our approach is based on a fine analysis of the Hausdorff density for $2$-varifolds with critical integrability of the mean curvature and a recent local regularity result by Bi-Zhou. In codimension one, there are only three possible density values below $2$, each of which can be attained with equality in the Li--Yau inequality for the Willmore functional by the unit sphere, the double bubble, and the triple bubble. We show that below an optimal threshold for the Willmore energy, a varifold induced by a current without boundary is in fact a curvature varifold with a uniform bound on its second fundamental form. Consequently, the minimization of the Willmore functional in the class of curvature varifolds with prescribed even Euler characteristic provides smooth solutions for the Willmore problem. In particular, the "ambient" varifold approach and the "parametric" approach are equivalent for minimizing the Willmore energy.