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 $\bar{M}={\mathbb{R}}^{n+k}$ 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 ${\mathbb{R}}^{n+k}$. 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 ${\mathbb{R}}^{n+k}$.

arXiv comment19 pages, 8 figures. This new version corrects the statement of Theorem 3 and fills a gap in the proof of Lemma 8, that now has changed completely. We have also added Theorem 9 which states that the approach to a shrinker torus by Angenent satisfies the hypotheses of Theorem 3. Moreover we have added an appendix with 8 new figures. There are also other minor corrections