Lord Rayleigh's Conjecture for Vibrating Clamped Plates in Positively Curved Spaces

TL;DR

This paper proves Lord Rayleigh's conjecture for vibrating clamped plates on positively curved Riemannian manifolds, extending known results to higher dimensions and revealing curvature's role in spectral properties.

Contribution

It provides the first positive resolution of Lord Rayleigh's conjecture in higher dimensions for positively curved spaces and introduces new spectral gap estimates and isoperimetric inequalities.

Findings

Affirmative proof of Lord Rayleigh's conjecture in 2 and 3 dimensions.

Extension of results to higher dimensions for large plates.

Establishment of a Lord Rayleigh-type isoperimetric inequality involving volume ratio.

Abstract

We affirmatively solve the analogue of Lord Rayleigh's conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (M. Ashbaugh & R. Benguria, 1995, and N. Nadirashvili, 1995) and negatively curved (A. Krist\'aly, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh's conjecture in higher dimensions. The proofs rely on an Ashbaugh-Benguria-Nadirashvili-Talenti nodal-decomposition argument, on the L\'evy-Gromov isoperimetric inequality, on fine properties of Gaussian hypergeometric functions and on sharp spectral gap estimates of fundamental tones for both small and large clamped spherical caps. Our results show that positive curvature enhances genuine differences between low- and high-dimensional settings, a tacitly accepted paradigm in the theory of vibrating clamped plates. In the limit case - when the Ricci curvature is non-negative - we establish a Lord Rayleigh-type isoperimetric inequality that involves the asymptotic volume ratio of the non-compact complete Riemannian manifold; moreover, the inequality is strongly rigid in 2 and 3 dimensions, i.e., if equality holds for a given clamped plate then the manifold is isometric to the Euclidean space.