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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.19086 |
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| _version_ | 1866914731670896640 |
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| author | Carron, Gilles Chen, Bo-Yong Xiong, Yuanpu |
| author_facet | Carron, Gilles Chen, Bo-Yong Xiong, Yuanpu |
| contents | In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρ<r\}$ on a complete Riemannian manifold $M$ as $r\rightarrow +\infty$, where $ρ$ is a Lipschitz continuous exhaustion function with $|\nablaρ|\leq1$ a.e. on $M$. We obtain several sharp results. First, if for all $r>r_0$ \[ r^2 λ_1(B_r)\ge γ>0, \] we obtain a sharp estimate of the volume growth: $|B_r|\ge cr^{μ(γ)}.$ Moreover when $γ>j_0^2\approx 5.784$, where $j_0$ denotes the first positive zero of the Bessel function $J_0$, then $M$ is hyperbolic and we have a Hardy type inequality. In the case where $r_0=0$, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that $Δρ^2\geq2μ(γ)>0$. In particular, when $\inf_MΔρ^2>4$, $M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19086 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Type problem, the first eigenvalue and Hardy inequalities Carron, Gilles Chen, Bo-Yong Xiong, Yuanpu Differential Geometry Analysis of PDEs In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρ<r\}$ on a complete Riemannian manifold $M$ as $r\rightarrow +\infty$, where $ρ$ is a Lipschitz continuous exhaustion function with $|\nablaρ|\leq1$ a.e. on $M$. We obtain several sharp results. First, if for all $r>r_0$ \[ r^2 λ_1(B_r)\ge γ>0, \] we obtain a sharp estimate of the volume growth: $|B_r|\ge cr^{μ(γ)}.$ Moreover when $γ>j_0^2\approx 5.784$, where $j_0$ denotes the first positive zero of the Bessel function $J_0$, then $M$ is hyperbolic and we have a Hardy type inequality. In the case where $r_0=0$, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that $Δρ^2\geq2μ(γ)>0$. In particular, when $\inf_MΔρ^2>4$, $M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied. |
| title | Type problem, the first eigenvalue and Hardy inequalities |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2403.19086 |