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

TL;DR

This paper derives a lower bound for the first eigenvalue of the Laplacian on hypersurfaces embedded in manifolds with positive Ricci curvature, extending classical minimal surface estimates to non-minimal cases.

Contribution

It provides a new eigenvalue estimate for hypersurfaces in positively curved manifolds, incorporating curvature bounds and geometric properties, generalizing previous minimal surface results.

Findings

Lower bound depends on ambient curvature bounds and hypersurface geometry

Extends classical eigenvalue estimates to non-minimal hypersurfaces

Applicable under specific Ricci and sectional curvature conditions

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.