Stability of eigenvalues and observable diameter in RCD(1,∞) spaces

2021/07/12 by Jérôme Bertrand, Bertrand, Jérôme, Max Fathi +1
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2107.05324

openalex publication_date 2021/07/12 · openalex created_date 2022/11/22 · openalex updated_date 2026/07/28

Abstract

We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1, ∞) 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.

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