An eigenvalue estimate for self-shrinkers in a Ricci shirinker

Franciele Conrado; Detang Zhou

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arXiv · 2505.03499

An eigenvalue estimate for self-shrinkers in a Ricci shirinker

Franciele Conrado·Detang Zhou

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Abstract

In this paper, we study the drifted Laplacian $\Delta_f$ on a hypersurface $M$ in a Ricci shrinker $(\overline{M},g,f)$. We prove that the spectrum of $\Delta_f$ 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 $\Delta_f$ 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.

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BibTeX RIS

Franciele Conrado, Detang Zhou. 2025-05-06. An eigenvalue estimate for self-shrinkers in a Ricci shirinker. https://arxiv.org/abs/2505.03499

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