Abstract In this paper, we study the drifted Laplacian Δ f on a hypersurface M in a Ricci shrinker (M ̄, g, f) (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.
Conrado et al. (2025) studied this question.