An eigenvalue estimate for self-shrinkers in a Ricci shirinker
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909602621161472 |
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| author | Conrado, Franciele Zhou, Detang |
| author_facet | Conrado, Franciele Zhou, Detang |
| contents | In this paper, we study the drifted Laplacian $Δ_f$ on a hypersurface $M$ in a Ricci shrinker $(\overline{M},g,f)$. We prove that the spectrum of $Δ_f$ is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of $Δ_f$ when the hypersurface is an embedded $f$-minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang's estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03499 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An eigenvalue estimate for self-shrinkers in a Ricci shirinker Conrado, Franciele Zhou, Detang Differential Geometry In this paper, we study the drifted Laplacian $Δ_f$ on a hypersurface $M$ in a Ricci shrinker $(\overline{M},g,f)$. We prove that the spectrum of $Δ_f$ is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of $Δ_f$ when the hypersurface is an embedded $f$-minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang's estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers. |
| title | An eigenvalue estimate for self-shrinkers in a Ricci shirinker |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2505.03499 |