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, sufficiently large dimensions beyond 3. These results complement those from flat (Ashbaugh Benguria Duke Math J 78(1):1–17, 1995; Nadirashvili Arch Ration Mech Anal 129(1):1–10, 1995) negatively curved (Kristály Adv 367:107113, 2020) cases that are valid only at same time also provide first answer to higher dimensions. The proofs rely an Ashbaugh–Benguria–Nadirashvili–Talenti nodal-decomposition argument, Lévy–Gromov isoperimetric inequality, fine properties Gaussian hypergeometric functions sharp spectral gap estimates fundamental tones both small spherical caps. Our show enhances genuine differences between low- high-dimensional settings, a tacitly accepted paradigm theory vibrating plates. In limit case—when is non-negative we establish Rayleigh-type inequality involves asymptotic volume ratio non-compact complete manifold; moreover, strongly rigid i.e., if equality holds given plate then manifold isometric Euclidean space.