First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds

We derive a sharp upper bound for the first eigenvalue ${\lambda }_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\mathrm{\infty }$. We then prove that a particular class of conformally compact submanifolds within asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold $Y^{k+1}$ within ${\mathbb{H}}^{n+1}(-1)$, ${\lambda }_{1,p}(Y)={\left(\frac{k}{p}\right)}^p$. We then obtain lower bounds on ${\lambda }_{1,2}(Y)$ in the case where minimality is replaced with a bounded mean curvature assumption and where the ambient space is a general Poincaré-Einstein space whose conformal infinity is of non-negative Yamabe type. In the process, we introduce an invariant ${\widehat{\beta }}^Y$ for each such submanifold, enabling us to generalize a result due to Cheung-Leung.