Sharp Spectral Bounds for the $p$-Laplacian and Polyharmonic Operators on Asymptotically Hyperbolic Manifolds

Abstract

We derive sharp bounds for three types of eigenvalue problems. First, we derive an upper bound for the first $p$-Dirichlet eigenvalue on conformally compact (CC) spaces. As a consequence, we show that for a class of CC submanifolds of asymptotically hyperbolic spaces, the asymptotic sectional curvatures, the meeting angle at infinity, and the vanishing of the norm of the mean curvature are all determined by its first $p$-Dirichlet eigenvalue. Additionally, we derive sharp upper bounds for the first eigenvalue of polyharmonic operators under both clamped and buckling boundary conditions. Finally, we prove sharp lower bounds for all three types of eigenvalue problems on weakly Poincaré-Einstein spaces with $\text{Ric}_{g_+}\ge -ng_+$ and whose conformal infinity has nonnegative Yamabe constant, and on their submanifolds.