Regularity for graphs with bounded anisotropic mean curvature

TL;DR

This paper proves regularity results for Lipschitz graphs with bounded anisotropic mean curvature, introduces a novel ellipticity condition, and provides new examples satisfying the atomic condition, advancing understanding in high codimension geometric measure theory.

Contribution

It establishes regularity for anisotropic mean curvature bounded graphs under a new ellipticity condition and constructs the first non-trivial examples satisfying the atomic condition in high codimension.

Findings

Regularity holds almost everywhere for graphs with bounded anisotropic mean curvature.

Introduces a new ellipticity condition that implies the atomic condition.

Provides the first non-trivial class of anisotropic energies satisfying the atomic condition in high codimension.

Abstract

We prove that $m$-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p>m$, are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a $C^2$ neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a $C^2$ (resp. $C^1$) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.