no. 6
Stability estimates for the sharp spectral gap bound under a curvature-dimension condition
Fathi, Max1; Gentil, Ivan2; Serres, Jordan3
1 Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques Louis Lions & Laboratoire de Probabilités Statistique et Modélisation, 8 place Aurélie Nemours, F-75013 Paris (France)
2 Institut Camille Jordan, Umr Cnrs 52065, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex (France)
3 CREST, ENSAE Paris, Institut Polytechnique de Paris 5 avenue Le Chatelier, 91120 Palaiseau (France)
Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2425-2459.

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.

Nous étudions la stabilité du trou spectral pour des espaces métriques mesurés vérifiant un critère de courbure dimension. Notre résultat principal, nouveau même dans le cadre d’un espace lisse, est une estimation quantitative montrant que si le trou spectral d’un espace RCD(N-1,N) est presque minimal alors l’image par la fonction propre associée au trou spectral de la mesure de référence est proche d’une loi Beta. La preuve combine des estimations de la fonction propre par le biais d’une nouvelle inégalité fonctionnelle de type L 1 dans un espace RCD et la méthode de Stein pour comparer des lois. Nous obtenons aussi des bornes presque optimales pour des espaces ayant une dimension intrinsèque infinie ou bien négative.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3608
Classification: 60B05, 46-XX
Keywords: Spectral gap, Poincaré inequalities, RCD spaces, Curvature-dimension condition, Stein method.
Mot clés : Trou spectral, inégalité de Poincaré, espaces RCD, condition de courbure-dimension, méthode de Stein.

Fathi, Max 1; Gentil, Ivan 2; Serres, Jordan 3

1 Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques Louis Lions & Laboratoire de Probabilités Statistique et Modélisation, 8 place Aurélie Nemours, F-75013 Paris (France)
2 Institut Camille Jordan, Umr Cnrs 52065, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex (France)
3 CREST, ENSAE Paris, Institut Polytechnique de Paris 5 avenue Le Chatelier, 91120 Palaiseau (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Fathi, Max; Gentil, Ivan; Serres, Jordan. Stability estimates for the sharp spectral gap bound under a curvature-dimension condition. Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2425-2459. doi : 10.5802/aif.3608. https://aif.centre-mersenne.org/articles/10.5802/aif.3608/

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