Submitted Paper

Inserted: 14 apr 2026
Last Updated: 14 apr 2026

Year: 2026

Abstract:

Suppose $ F $ is an integrand associated with a uniformly convex $ 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 {\rm spt} \
V \
$ is absolutely continuous with respect to $ \
V \
$ and the mean $ F $-curvature $ h_F(V, \cdot) $ is bounded in $ L^\infty $. In a previous result we prove that $ {\rm spt} \
V \
$ is $ C^2 $-rectifiable and the $ C^1$-regular part $ M $ of $ {\rm spt} \
V \
$ coincides $ \mathscr{H}^n $ almost everywhere with the unit-density stratum of $ V $. In this paper we prove that $ h_F(V,a) \in {\rm Nor}(M,a) $ for $ \mathscr{H}^n $ a.e. $ a \in M $ and that $ h_F(V, \cdot) $ agrees with the approximate mean $ F $-curvature coming from the $ 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.