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

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.