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

TL;DR

This paper extends the isoperimetric inequality for the principal frequency of clamped plates to RCD(0,N) metric measure spaces, establishing sharpness, rigidity, and stability results even in singular settings.

Contribution

It introduces a new isoperimetric inequality for the principal frequency in RCD(0,N) spaces near N=2 or 3, including rigidity and stability analyses.

Findings

Inequality contains asymptotic volume ratio

Results are sharp under subharmonicity of the distance function

Rigidity and stability are characterized by cone structure and eigenfunction shape

Abstract

We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature and dimension bounded above by N>1 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., 1995] and Nadirashvili [Arch. Rat. Mech. Anal., 1995]. The main contribution of the present work is a new isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces whenever N 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 RCD(0,N) 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 non-negative Ricci curvature. We discuss examples of both smooth and non-smooth spaces where the results can be applied.