Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914315059068928 |
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| author | Benatti, Luca Pluda, Alessandra Pozzetta, Marco |
| author_facet | Benatti, Luca Pluda, Alessandra Pozzetta, Marco |
| contents | We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In $3$-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of $p$-capacitary potentials. We prove that they strongly converge in $W^{1,q}_{\mathrm{loc}}$ as $p\to 1^+$ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas Benatti, Luca Pluda, Alessandra Pozzetta, Marco Differential Geometry Analysis of PDEs Functional Analysis 53C21, 53E10, 38C35, 49J45 We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In $3$-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of $p$-capacitary potentials. We prove that they strongly converge in $W^{1,q}_{\mathrm{loc}}$ as $p\to 1^+$ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials. |
| title | Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas |
| topic | Differential Geometry Analysis of PDEs Functional Analysis 53C21, 53E10, 38C35, 49J45 |
| url | https://arxiv.org/abs/2411.06462 |