First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds

We derive a sharp upper bound for the first eigenvalue ₁, ₏ of the p-Laplacian on asymptotically hyperbolic manifolds for 1<p<. 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 H^n+1 (-1), ₁, ₏ (Y) = (kp) ^p. We then obtain lower bounds on 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 Y for each such submanifold, enabling us to generalize a result due to Cheung-Leung.