"Universal" inequalities for the eigenvalues of the biharmonic operator

TL;DR

This paper derives universal inequalities for eigenvalues of the biharmonic operator across various geometric settings, including Euclidean spaces, spheres, projective spaces, and hyperbolic spaces, extending previous spectral bounds.

Contribution

It introduces new universal eigenvalue inequalities for the biharmonic operator on diverse Riemannian manifolds and submanifolds, broadening the scope of spectral geometry results.

Findings

Established inequalities for eigenvalues on compact submanifolds of Euclidean spaces

Extended inequalities to spheres, projective spaces, and hyperbolic spaces

Applicable to domains with spherical eigenmaps and homogeneous Riemannian spaces

Abstract

In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherical eigenmaps (this includes the compact homogeneous Riemannian spaces) and nally on domains of the hyperbolic space.