  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.
