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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.21983 |
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| _version_ | 1866911593440215040 |
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| author | Kolasiński, Sławomir Santilli, Mario |
| author_facet | Kolasiński, Sławomir Santilli, Mario |
| contents | 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, Schätzle and Ambrosio-Masnou. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature Kolasiński, Sławomir Santilli, Mario Analysis of PDEs Differential Geometry Optimization and Control 53A35, 49Q20, 49Q15 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, Schätzle and Ambrosio-Masnou. |
| title | Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature |
| topic | Analysis of PDEs Differential Geometry Optimization and Control 53A35, 49Q20, 49Q15 |
| url | https://arxiv.org/abs/2603.21983 |