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 $1<p<\infty$. 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 Poincar\'e-Einstein space whose conformal infinity is of non-negative Yamabe type. In the process, we introduce an invariant $\hat \beta^Y$ for each such submanifold, enabling us to generalize a result due to Cheung-Leung.