Allard's interior $\varepsilon$-Regularity Theorem in Alexandrov spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866908321765654528 |
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| author | Agnoletto, Marcos Hoyos, Julio C. Correa da Silva, Márcio Fabiano Nardulli, Stefano |
| author_facet | Agnoletto, Marcos Hoyos, Julio C. Correa da Silva, Márcio Fabiano Nardulli, Stefano |
| contents | In this paper, we prove Allard's Interior $\varepsilon$-Regularity Theorem for $m$-dimensional varifolds with generalized mean curvature in $L^p_{loc}$, for $p \in \mathbb{R}$ such that $p>m$, in Alexandrov spaces of dimension $n$ with double-sided bounded intrinsic sectional curvature. We first give an intrinsic proof of this theorem in the case of varifolds in Riemannian manifolds of dimension $n$ whose metric tensor is at least of class $\mathcal{C}^2$, without using Nash's Isometric Embedding Theorem. This approach provides explicitly computable constants that depend only on $n$, $m$, the injectivity radius and bounds on the sectional curvature, which is essential for proving our main theorem, as we establish it through a density argument in the topological space of Riemannian manifolds with positive lower bounds on the injectivity radius and double-sided bounds on sectional curvature, equipped with the $\mathcal{C}^{1,α}$ topology, for every $α\in ]0,1[$ (in fact, it is enough with the $W^{2,q}$ topology for some suitable $q$ large enough). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10758 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Allard's interior $\varepsilon$-Regularity Theorem in Alexandrov spaces Agnoletto, Marcos Hoyos, Julio C. Correa da Silva, Márcio Fabiano Nardulli, Stefano Differential Geometry Analysis of PDEs Metric Geometry 49Q20, 49Q15, 58A25, 58C35, 28A75, 53A10 In this paper, we prove Allard's Interior $\varepsilon$-Regularity Theorem for $m$-dimensional varifolds with generalized mean curvature in $L^p_{loc}$, for $p \in \mathbb{R}$ such that $p>m$, in Alexandrov spaces of dimension $n$ with double-sided bounded intrinsic sectional curvature. We first give an intrinsic proof of this theorem in the case of varifolds in Riemannian manifolds of dimension $n$ whose metric tensor is at least of class $\mathcal{C}^2$, without using Nash's Isometric Embedding Theorem. This approach provides explicitly computable constants that depend only on $n$, $m$, the injectivity radius and bounds on the sectional curvature, which is essential for proving our main theorem, as we establish it through a density argument in the topological space of Riemannian manifolds with positive lower bounds on the injectivity radius and double-sided bounds on sectional curvature, equipped with the $\mathcal{C}^{1,α}$ topology, for every $α\in ]0,1[$ (in fact, it is enough with the $W^{2,q}$ topology for some suitable $q$ large enough). |
| title | Allard's interior $\varepsilon$-Regularity Theorem in Alexandrov spaces |
| topic | Differential Geometry Analysis of PDEs Metric Geometry 49Q20, 49Q15, 58A25, 58C35, 28A75, 53A10 |
| url | https://arxiv.org/abs/2504.10758 |