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

Abstract: Let $(\bar{M},<,>,e^\psi)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold of $\bar{M}$ with the induced metric and density. We give an upper bound on the first eigenvalue $\lambda_1$ of the closed eigenvalue problem for $\Delta_\psi$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector ${\vec{H}}{{\psi}} + {\bar \nabla}$ with respect to the volume form induced by the density, where ${\vec{H}}{{\psi}}$ is the mean curvature of $M$ associated to the density $e^\psi$. When $\bar{M}=\Bbb R^{n+k}$ or $\bar{M}=S^{n+k-1}$, the equality between $\lambda_1$ and its bound implies that $e^\psi$ is a Gaussian density ($\psi(x) = \frac{C}{2} |x|^2$, $C<0$), and $M$ is a shrinker for the mean curvature flow (MCF) on $\Bbb R^{n+k}$. We prove also that $\lambda_1 =-C$ on the standard shrinker torus of revolution. Based on this and on the Yau's conjecture on the first eigenvalue of minimal submanifolds of $S^n$, we conjecture that the equality $\lambda_1=-C$ is true for all the shrinkers of MCF in $\mathbb{R}^{n+k}$.