A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909979269660672 |
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| author | Li, Fagui Yan, Junrong |
| author_facet | Li, Fagui Yan, Junrong |
| contents | Let $Σ$ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold $N$. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of $N$, we establish a lower bound for the first nonzero eigenvalue of the Laplacian on $Σ$. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of $Σ$ through its mean curvature and second fundamental form. This result extends the classical eigenvalue estimate of Choi and Wang [J. Diff. Geom. \textbf{18} (1983), 559--562.] to the non-minimal case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02803 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds Li, Fagui Yan, Junrong Differential Geometry 58C40, 58J50 Let $Σ$ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold $N$. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of $N$, we establish a lower bound for the first nonzero eigenvalue of the Laplacian on $Σ$. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of $Σ$ through its mean curvature and second fundamental form. This result extends the classical eigenvalue estimate of Choi and Wang [J. Diff. Geom. \textbf{18} (1983), 559--562.] to the non-minimal case. |
| title | A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds |
| topic | Differential Geometry 58C40, 58J50 |
| url | https://arxiv.org/abs/2308.02803 |