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

TL;DR

This paper establishes precise spectral bounds for eigenvalues of the $p$-Laplacian and polyharmonic operators on asymptotically hyperbolic manifolds, linking geometric properties to spectral data.

Contribution

It provides new sharp bounds for eigenvalues on conformally compact spaces and submanifolds, connecting geometric invariants with spectral estimates.

Findings

Sharp upper bounds for first $p$-Dirichlet eigenvalues on conformally compact spaces.

Eigenvalues determine asymptotic geometric features like curvature and meeting angles.

Sharp bounds for polyharmonic eigenvalues under various boundary conditions.

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 Poincar\'e-Einstein spaces with $\text{Ric}_{g_+}\ge -ng_+$ and whose conformal infinity has nonnegative Yamabe constant, and on their submanifolds.