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arXiv · 2308.02803

A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds

Fagui Li·Junrong Yan
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Abstract

Let $\Sigma$ 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 $\Sigma$. The estimate depends on the ambient curvature bounds, the normal injectivity radius, and the geometry of $\Sigma$ 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.

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Fagui Li, Junrong Yan. 2023-08-05. A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds. https://arxiv.org/abs/2308.02803

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