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

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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal-cnrs.archives-ouvertes.fr/hal-03845818
Contributor : Max Fathi Connect in order to contact the contributor
Submitted on : Wednesday, November 9, 2022 - 5:43:33 PM
Last modification on : Thursday, November 10, 2022 - 4:33:47 AM

### Identifiers

• HAL Id : hal-03845818, version 1
• ARXIV : 2202.03769

### Citation

Max Fathi, Ivan Gentil, Jordan Serres. Stability estimates for the sharp spectral gap bound under a curvature-dimension condition. 2022. ⟨hal-03845818⟩

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