Paper detail

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

Let $(\bar{M},<,>,e^ψ)$ 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 $λ_1$ of the closed eigenvalue problem for $Δ_ψ$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector ${\vec{H}}_{ψ} + {\bar \nabla}$ with respect to the volume form induced by the density, where ${\vec{H}}_{ψ}$ is the mean curvature of $M$ associated to the density $e^ψ$. When $\bar{M}=\Bbb R^{n+k}$ or $\bar{M}=S^{n+k-1}$, the equality between $λ_1$ and its bound implies that $e^ψ$ is a Gaussian density ($ψ(x) = \frac{C}{2} |x|^2$, $C<0$), and $M$ is a shrinker for the mean curvature flow (MCF) on $\Bbb R^{n+k}$. We prove also that $λ_1 =-C$ 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 $λ_1=-C$ is true for all the shrinkers of MCF in $\mathbb{R}^{n+k}$.