Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF

Let (M‾,〈,〉,eψ) be a Riemannian manifold with density, and let M closed n-dimensional submanifold of M‾ the induced metric density. We give an upper bound on first eigenvalue λ1 problem for Δψ (the Laplacian associated to density) in terms average norm vector H→ψ+∇‾ψ respect volume form by where H→ψ is mean curvature density eψ. When M‾=Rn+k or M‾=Sn+k−1, equality between its implies that eψ Gaussian (ψ(x)=C2|x|2, C<0), shrinker flow (MCF) Rn+k. prove also λ1=−C standard torus revolution. Based this Yau's conjecture minimal submanifolds Sn, we true all shrinkers MCF