Stability of Eigenvalues and Observable Diameter in RCD$$(1, \infty )$$ Spaces

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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