First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds

We derive a sharp upper bound for the first eigenvalue λ1,p 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 Yᵏ⁺¹ within Hⁿ⁺¹(-1), λ1,p(Y)=(k/p)ᵖ. We then obtain lower bounds on λ1,2(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 βY for each such submanifold, enabling us to generalize a result due to Cheung-Leung.