Stability estimates for the sharp spectral gap bound under a curvature-dimension condition

TL;DR

This paper establishes a quantitative stability estimate for the spectral gap in metric-measure spaces with curvature bounds, showing near-minimal spectral gaps imply the measure's pushforward is close to a Beta distribution, using new inequalities and Stein's method.

Contribution

It provides the first sharp stability estimate for spectral gaps in RCD spaces, extending results to infinite and negative dimensions, and introduces new $L^1$ inequalities and distribution approximation techniques.

Findings

Near-minimal spectral gap implies measure pushforward close to Beta distribution.

Develops new $L^1$-functional inequalities for RCD spaces.

Extends stability estimates to infinite and negative dimensions.

Abstract

We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing that if the spectral gap of an RCD$(N-1, N)$ space is almost minimal, then the pushforward of the measure by an eigenfunction associated with the spectral gap is close to a Beta distribution. The proof combines estimates on the eigenfunction obtained via a new $L^1$-functional inequality for RCD spaces with Stein's method for distribution approximation. We also derive analogous, almost sharp, estimates for infinite and negative values of the dimension parameter.