New Asymptotic Geometric Quantities in Riemannian Geometry and their Geometric applications
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913173387345920 |
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| author | Jin, Xiaoshang Yin, Jiabin |
| author_facet | Jin, Xiaoshang Yin, Jiabin |
| contents | This paper studies the large $p$ asymptotics of three geometric quantities on complete noncompact Riemannian manifolds: the $p-$capacity of a compact set, the first Dirichlet $p-$eigenvalue, and the Maz'ya constant, thereby offering a new perspective on the study of such manifolds. We introduce the infinity capacity $\mathcal{C}(Ω)$, the infinity eigenvalue $Λ(M)$, and the Maz'ya limit $\mathcal{M}(M)$, and establish the general inequality, for any $Ω\subset M$, $$ \mathcal{V}(M) \ge \mathcal{C}(Ω) \ge Λ(M) = \mathcal{M}(M), $$ where $\mathcal{V}(M)$ is the volume entropy. Under geometric conditions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, these quantities coincide and equal $\mathcal{V}(M)$ or the dimension. Finally, combining with the entropy rigidity theorem, we obtain a characterization of hyperbolic manifolds. We also provide examples showing strict inequalities hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14600 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | New Asymptotic Geometric Quantities in Riemannian Geometry and their Geometric applications Jin, Xiaoshang Yin, Jiabin Differential Geometry This paper studies the large $p$ asymptotics of three geometric quantities on complete noncompact Riemannian manifolds: the $p-$capacity of a compact set, the first Dirichlet $p-$eigenvalue, and the Maz'ya constant, thereby offering a new perspective on the study of such manifolds. We introduce the infinity capacity $\mathcal{C}(Ω)$, the infinity eigenvalue $Λ(M)$, and the Maz'ya limit $\mathcal{M}(M)$, and establish the general inequality, for any $Ω\subset M$, $$ \mathcal{V}(M) \ge \mathcal{C}(Ω) \ge Λ(M) = \mathcal{M}(M), $$ where $\mathcal{V}(M)$ is the volume entropy. Under geometric conditions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, these quantities coincide and equal $\mathcal{V}(M)$ or the dimension. Finally, combining with the entropy rigidity theorem, we obtain a characterization of hyperbolic manifolds. We also provide examples showing strict inequalities hold. |
| title | New Asymptotic Geometric Quantities in Riemannian Geometry and their Geometric applications |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2604.14600 |