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arXiv · 2107.05324

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

Jérôme Bertrand·Max Fathi
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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.

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Jérôme Bertrand, Max Fathi. 2021-07-12. Stability of eigenvalues and observable diameter in RCD$(1,\infty)$ spaces. https://arxiv.org/abs/2107.05324

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