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 (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.