Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds

TL;DR

This paper establishes sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds, using elementary functional inequalities, and addresses open questions about eigenvalue behavior in hyperbolic spaces.

Contribution

It provides the first symmetrization-free proof of sharp spectral gap estimates for higher-order operators on these manifolds, solving open problems and deriving related inequalities.

Findings

Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds.

Resolution of a conjecture on eigenvalue asymptotics in hyperbolic spaces.

Derivation of various Rellich inequalities in the same geometric setting.

Abstract

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan-Hadamard manifolds. The proofs are symmetrization-free -- thus no sharp isoperimetric inequality is needed -- based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from Cheng and Yang [Proc. Amer. Math. Soc., 2011] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in Krist\'aly [Adv. Math., 2020] on the validity of such sharp estimates in high-dimensional Cartan-Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.