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

Alexandru Kristály, Andrea Mondino

TL;DR

This work extends Rayleigh’s isoperimetric inequality for the principal frequency of clamped plates to the synthetic setting of $RCD(0,N)$ spaces, proving a sharp bound for dimensions near $N\in(2-\varepsilon_0,2+\varepsilon_0)\cup(3-\varepsilon_0,3+\varepsilon_0)$ that incorporates the asymptotic volume ratio $AVR_{\sf m}$. The authors reduce the problem to a one-dimensional model via level-set rearrangements, co-area, and Gauss–Green formulas, and then analyze a Bessel-function–based 1D variational problem in the Spirit of Ashbaugh–Benguria to obtain the main inequality. They establish sharpness and rigidity: equality is attained by metric cones, and equality forces the ambient space to be an $N$-Euclidean cone with the domain a ball centered at the cone tip; eigenfunctions have explicit conical–Bessel forms. A stability theory is developed showing that near-optimal configurations are pmGH-close to model cones and that near-extremal eigenfunctions approximate the model profile, with broad applicability to smooth and non-smooth spaces. Overall, the paper extends classical spectral isoperimetric results to non-smooth curvature-dimension spaces and highlights the decisive role of conic geometry and $AVR_{\sf m}$ in higher-order eigenvalue problems.

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.

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

TL;DR

This work extends Rayleigh’s isoperimetric inequality for the principal frequency of clamped plates to the synthetic setting of spaces, proving a sharp bound for dimensions near that incorporates the asymptotic volume ratio . The authors reduce the problem to a one-dimensional model via level-set rearrangements, co-area, and Gauss–Green formulas, and then analyze a Bessel-function–based 1D variational problem in the Spirit of Ashbaugh–Benguria to obtain the main inequality. They establish sharpness and rigidity: equality is attained by metric cones, and equality forces the ambient space to be an -Euclidean cone with the domain a ball centered at the cone tip; eigenfunctions have explicit conical–Bessel forms. A stability theory is developed showing that near-optimal configurations are pmGH-close to model cones and that near-extremal eigenfunctions approximate the model profile, with broad applicability to smooth and non-smooth spaces. Overall, the paper extends classical spectral isoperimetric results to non-smooth curvature-dimension spaces and highlights the decisive role of conic geometry and in higher-order eigenvalue problems.

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.
Paper Structure (16 sections, 13 theorems, 171 equations)

This paper contains 16 sections, 13 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev spaces on ${\sf RCD}(0,N)$ spaces
  4. Rearrangements.
  5. Sharp isoperimetric inequality on ${\sf RCD}(0,N)$ spaces
  6. Coupled minimization problem: from ${\sf RCD}(0,N)$ to the model space
  7. Fine level-set analysis
  8. Nodal decomposition and the appearance of ${\sf AVR}_{\sf m}$
  9. Reducing the coupled minimization: proof of Theorem \ref{['theorem-main']}
  10. Ashbaugh--Benguria's argument in the 1-dimensional model space
  11. Concluding the proof of inequality \ref{['inequality']}
  12. Sharpness and rigidity in the main inequality \ref{['inequality']}: proof of Theorem \ref{['theorem-main-sharpness-rigidity']}
  13. Sharpness: proof of Theorem \ref{['theorem-main-sharpness-rigidity']}/(i)
  14. Rigidity: proof of Theorem \ref{['theorem-main-sharpness-rigidity']}/(ii)
  15. Stability in \ref{['inequality']}: proof of Theorem \ref{['theorem-main-stability']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Isoperimetric inequality for clamped plates on ${\sf RCD}(0,N)$ spaces) There exists $\varepsilon_0>0$ with the following property. Let $N\in (2-\varepsilon_0,2+\varepsilon_0)\cup (3-\varepsilon_0,3+\varepsilon_0)$ and let $(X,{\sf d},{\sf m})$ be an ${\sf RCD}(0,N)$ metric measure space with ${\sf where $\Omega^*=[0,R]$ with ${\sf m}(\Omega)=\omega_N R^N$, and $\Lambda_0(\Omega^*)$ stands for th

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 14 more