Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

TL;DR

This paper extends classical Euclidean theorems to anisotropic settings, showing that bounded anisotropic mean curvature varifolds have rectifiable supports and curvature vectors aligned with the normal, under convexity conditions.

Contribution

It proves that anisotropic mean curvature vectors lie in the normal bundle and match the approximate curvature, generalizing Euclidean results to anisotropic varifolds.

Findings

Mean curvature vector is in the normal bundle almost everywhere.

Curvature agrees with the approximate mean curvature from the rectifiable covering.

Supports are proven to be $\mathscr{C}^{2}$-rectifiable.

Abstract

Suppose $ F $ is an integrand associated with a uniformly convex $ \mathscr{C}^{3} $-norm, and $ V $ is a $ n $-dimensional varifold in an open subset of $ \mathbf{R}^{n+1} $ such that $ \mathscr{H}^n \llcorner \operatorname{spt} \| V \| $ is absolutely continuous with respect to $ \| V \| $ and the mean $ F $-curvature $ \mathbf{h}_{F}(V, \cdot) $ is bounded in $\mathbf{L}^\infty $. In our previous result arXiv:2507.18357 we prove that $ \operatorname{spt} \| V \| $ is $ \mathscr{C}^{2} $-rectifiable and the $ \mathscr{C}^{1} $-regular part $ M $ of $ \operatorname{spt} \| V \| $ coincides $ \mathscr{H}^n $ almost everywhere with the unit-density stratum of $ V $. In this paper we prove that $ \mathbf{h}_{F}(V,a) \in \operatorname{Nor}(M,a) $ for $ \mathscr{H}^n $ a.e.\ $ a \in M $ and that $ \mathbf{h}_{F}(V, \cdot) $ agrees with the approximate mean $ F $-curvature coming from the $ \mathscr{C}^{2} $-rectifiable covering of $ M $. These results provide anisotropic extensions of well known theorems in the Euclidean setting by Brakke, Sch\"atzle and Ambrosio-Masnou.