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

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 (Ashbaugh and Benguria in Duke Math J 78(1):1–17, 1995; Nadirashvili in Arch Ration Mech Anal 129(1):1–10, 1995) and negatively curved (Kristály in Adv Math 367:107113, 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 Lévy–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.