Principal frequency of clamped plates on RCD(0,N)RCD(0,N) spaces: Sharpness, rigidity, and stability

We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the condition, that is, infinitesimally Hilbertian spaces with nonnegative Ricci curvature and dimension bounded above by in the synthetic sense. The initial conjecture, an isoperimetric inequality for the principal frequency of clamped plates, has been formulated in 1877 by Lord Rayleigh in the Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and Benguria ( Duke Math. J . 78 (1) (1995), 1–17) and Nadirashvili ( Arch. Rat. Mech. Anal . 129 (1) (1995), 1–10). The main contribution of the present work is a new isoperimetric inequality for the principal frequency of clamped plates in spaces whenever is close enough to 2 or 3. The inequality contains the so‐called asymptotic volume ratio , and turns out to be sharp under the subharmonicity of the distance function, a condition satisfied in metric measure cones. In addition, rigidity (i.e., equality in the isoperimetric inequality) and stability results are established in terms of the cone structure of the space as well as the shape of the eigenfunction for the principal frequency, given by means of Bessel functions. These results are new even for Riemannian manifolds with nonnegative Ricci curvature. We discuss examples of both smooth and nonsmooth spaces where the results can be applied.