First eigenvalue estimates for asymptotically hyperbolic manifolds and their submanifolds

Abstract

We derive a sharp upper bound for the first eigenvalue ${\lambda }_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic (AH) manifolds for $1<p<\infty$. We show that asymptotically constant mean curvature submanifolds within AH manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold $Y^{k+1}$ within ${\mathbb{H}}^{n+1}(-1)$, the first eigenvalue ${\lambda }_{1,p}(Y)$ satisfies ${\lambda }_{1,p}(Y)=(k/p{)}^p$. Finally, we obtain lower bounds for ${\lambda }_{1,2}(Y)$ for complete, non-compact submanifolds with bounded mean curvature in a large class of AH spaces. In the course of this analysis, we introduce an invariant ${\hat{\beta }}^Y$ for each such submanifold.