Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917211870855168 |
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| author | Jin, Xiaoshang Lü, Zhiwei |
| author_facet | Jin, Xiaoshang Lü, Zhiwei |
| contents | We establish a lower bound for the principal $p-$frequency $λ_{1,p}(Ω)$ on a bounded domain $Ω$ in a non-compact Riemannian manifold of dimension $n.$ Under the assumption that the Ricci curvature satisfies $\operatorname{Ric} \geq (n-1)K$ with $K<0,$ we prove that $λ_{1,p}(Ω) > \barλ_{D,K,n}$, where $D$ is the diameter of $Ω$ and $\barλ_{D,K,n}$ is explicitly defined as the first eigenvalue of an associated one-dimensional ordinary differential equation model that incorporates both $D$ and $K.$ Moreover, the estimate is sharp. This work extends previous results for the case $K=0$ to the geometrically more complex setting of negative Ricci curvature, and providing a new quantitative connection between the eigenvalue, the diameter of domains, and the curvature lower bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14018 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature Jin, Xiaoshang Lü, Zhiwei Differential Geometry Spectral Theory We establish a lower bound for the principal $p-$frequency $λ_{1,p}(Ω)$ on a bounded domain $Ω$ in a non-compact Riemannian manifold of dimension $n.$ Under the assumption that the Ricci curvature satisfies $\operatorname{Ric} \geq (n-1)K$ with $K<0,$ we prove that $λ_{1,p}(Ω) > \barλ_{D,K,n}$, where $D$ is the diameter of $Ω$ and $\barλ_{D,K,n}$ is explicitly defined as the first eigenvalue of an associated one-dimensional ordinary differential equation model that incorporates both $D$ and $K.$ Moreover, the estimate is sharp. This work extends previous results for the case $K=0$ to the geometrically more complex setting of negative Ricci curvature, and providing a new quantitative connection between the eigenvalue, the diameter of domains, and the curvature lower bound. |
| title | Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature |
| topic | Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2601.14018 |