An eigenvalue estimate for self-shrinkers in a Ricci shirinker

TL;DR

This work analyzes the drifted Laplacian $Δ_f$ on hypersurfaces within a Ricci shrinker, proving discreteness of its spectrum for properly immersed $f$-minimal (or CWMC) hypersurfaces under mild potential-growth conditions and establishing a universal lower bound $λ_1(Δ_f) ≥ \frac{k}{2}$ (with $k$ normalized to $\tfrac{1}{2}$ giving $λ_1 ≥ \tfrac{1}{4}$). It uses a Reilly-type weighted Bochner inequality together with an eigenfunction-extension argument to derive sharp, dimension-independent lower bounds that recover classical results for compact minimal hypersurfaces and recent shrinker bounds by Ding–Xin and Brendle–Tsiamis. The results unify spectral discreteness and eigenvalue estimates for noncompact shrinker geometries, highlighting the role of the potential $f$ and the weighted measure $e^{-f} dv$ in controlling the spectrum. They provide a robust framework for spectral theory of drifted Laplacians on noncompact, curved ambient spaces relevant to mean curvature flow and CWMC geometry.

Abstract

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.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.1
• Theorem 1.6
• Remark 1.1
• Remark 1.2
• Theorem 2.1: Chen, chen