An eigenvalue estimate for self-shrinkers in a Ricci shirinker

TL;DR

This paper investigates the spectral properties of the drifted Laplacian on hypersurfaces within Ricci shrinkers, establishing discreteness of the spectrum and providing lower bounds for the first nonzero eigenvalue, extending known results in minimal hypersurface theory.

Contribution

It introduces new eigenvalue estimates for the drifted Laplacian on hypersurfaces in Ricci shrinkers, generalizing previous results for minimal hypersurfaces in spheres.

Findings

Spectrum of $ riangle_f$ is discrete under certain conditions.

Lower bounds for the first nonzero eigenvalue of $ riangle_f$ are established.

Results recover and extend known estimates for self-shrinkers and minimal hypersurfaces.

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.