An eigenvalue estimate for self-shrinkers in a Ricci shrinker

In this paper, we study the drifted Laplacian Δ f on a hypersurface M in a Ricci shrinker <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M̄,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.