First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909301889564672 |
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| author | Pérez-Ayala, Samuel Tyrrell, Aaron J. |
| author_facet | Pérez-Ayala, Samuel Tyrrell, Aaron J. |
| contents | We derive a sharp upper bound for the first eigenvalue $λ_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold $Y^{k+1}$ within $\mathbb{H}^{n+1}(-1)$, $λ_{1,p}(Y)=\left(\frac{k}{p}\right)^{p}$. We then obtain lower bounds on $λ_{1,2}(Y)$ in the case where minimality is replaced with a bounded mean curvature assumption and where the ambient space is a general Poincaré-Einstein space whose conformal infinity is of non-negative Yamabe type. In the process, we introduce an invariant $\hat β^Y$ for each such submanifold, enabling us to generalize a result due to Cheung-Leung. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds Pérez-Ayala, Samuel Tyrrell, Aaron J. Differential Geometry Spectral Theory We derive a sharp upper bound for the first eigenvalue $λ_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold $Y^{k+1}$ within $\mathbb{H}^{n+1}(-1)$, $λ_{1,p}(Y)=\left(\frac{k}{p}\right)^{p}$. We then obtain lower bounds on $λ_{1,2}(Y)$ in the case where minimality is replaced with a bounded mean curvature assumption and where the ambient space is a general Poincaré-Einstein space whose conformal infinity is of non-negative Yamabe type. In the process, we introduce an invariant $\hat β^Y$ for each such submanifold, enabling us to generalize a result due to Cheung-Leung. |
| title | First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds |
| topic | Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2404.07365 |