Type problem and the first eigenvalue
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911762312331264 |
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| author | Chen, Bo-Yong Xiong, Yuanpu |
| author_facet | Chen, Bo-Yong Xiong, Yuanpu |
| contents | In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρ<r\}$ on a complete Riemannian manfold $M$ as $r\rightarrow +\infty$, where $ρ$ is a Lipschitz continuous exhaustion function with $|\nablaρ|\leq1$ a.e. on $M$. We show that $M$ is hyperbolic whenever \[ Λ_*:= \liminf_{r\rightarrow +\infty} \{ r^2 λ_1(B_r)\} >18.624\cdots. \] Moreover, an upper bound of $Λ_*$ in terms of volume growth $ν_*:=\liminf_{r\rightarrow +\infty} \frac{\log |B_r|}{\log r}$ is given as follows \[ {Λ_*} \lesssim \begin{cases} ν_*^2,\ \ \ &ν_*\gg1,\\ ν_*\log\frac{1}{ν_*},&1<ν_*\ll1. \end{cases} \] The exponent $2$ for $ν_*\gg1$ turns out to be the best possible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_11803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Type problem and the first eigenvalue Chen, Bo-Yong Xiong, Yuanpu Differential Geometry Complex Variables In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρ<r\}$ on a complete Riemannian manfold $M$ as $r\rightarrow +\infty$, where $ρ$ is a Lipschitz continuous exhaustion function with $|\nablaρ|\leq1$ a.e. on $M$. We show that $M$ is hyperbolic whenever \[ Λ_*:= \liminf_{r\rightarrow +\infty} \{ r^2 λ_1(B_r)\} >18.624\cdots. \] Moreover, an upper bound of $Λ_*$ in terms of volume growth $ν_*:=\liminf_{r\rightarrow +\infty} \frac{\log |B_r|}{\log r}$ is given as follows \[ {Λ_*} \lesssim \begin{cases} ν_*^2,\ \ \ &ν_*\gg1,\\ ν_*\log\frac{1}{ν_*},&1<ν_*\ll1. \end{cases} \] The exponent $2$ for $ν_*\gg1$ turns out to be the best possible. |
| title | Type problem and the first eigenvalue |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2401.11803 |