Cheng's eigenvalue comparison on metric measure spaces and applications

TL;DR

This paper extends Cheng's eigenvalue comparison to non-smooth metric measure spaces with Ricci curvature bounds, providing sharp bounds, rigidity, and stability results with applications in mathematics and physics.

Contribution

It generalizes Cheng's eigenvalue bounds to non-smooth spaces satisfying Ricci curvature conditions, including stability and rigidity results, and applies these to spectral bounds in physics.

Findings

Sharp upper bounds on first Dirichlet eigenvalues in non-smooth spaces

Rigidity and stability results for RCD* spaces

Bounds on Neumann eigenvalues and spectrum in metric measure spaces

Abstract

Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth setting of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense, via optimal transport. Rigidity and stability statements are provided for $\mathsf{RCD}^{\star}(K,N)$ spaces; the stability seems to be new even for smooth Riemannian manifolds. We then present some mathematical and physical applications: in the former, we obtain an upper bound on the $j^{th}$ Neumann eigenvalue in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces and a bound on the essential spectrum in non-compact $\mathsf{RCD}^{\star}(K,N)$ spaces; in the latter, the eigenvalue bounds correspond to general upper bounds on the masses of the spin-2 Kaluza-Klein excitations around general warped compactifications of higher-dimensional theories of gravity.