Stability of eigenvalues and observable diameter in RCD$(1,\infty)$ spaces

TL;DR

This paper investigates the stability of spectral gap and observable diameter in RCD(1,∞) metric measure spaces, establishing quantitative bounds and characterizing spaces close to Gaussian components.

Contribution

It provides new stability results linking spectral gap, observable diameter, and Gaussian components in RCD spaces with explicit bounds.

Findings

Spaces with almost maximal spectral gap contain near-Gaussian components.

Eigenvalues of the Laplacian are close to integers with dimension-free bounds.

Maximal spectral gap is equivalent to maximal observable diameter under certain conditions.

Abstract

We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1, $\infty$) condition. We show that if such a space has an almost maximal spectral gap, then it almost contains a Gaussian component, and the Laplacian has eigenvalues that are close to any integers, with dimension-free quantitative bounds. Under the additional assumption that the space admits a needle disintegration, we show that the spectral gap is almost maximal iff the observable diameter is almost maximal, again with quantitative dimension-free bounds.