A rigidity result for metric measure spaces with Euclidean heat kernel

Carron, Gilles; Tewodrose, David

doi:10.5802/jep.179

A rigidity result for metric measure spaces with Euclidean heat kernel
[Un résultat de rigidité pour les espaces métriques mesurés à noyau de la chaleur euclidien]

Carron, Gilles ¹ ; Tewodrose, David ²

¹ Université de Nantes, Département de Mathématiques 2 rue de la Houssinière, BP 92208, 44322 Nantes cedex 03, France
² CY Cergy Paris University, AGM 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 101-154.

Résumé
Abstract

Nous prouvons qu’un espace métrique mesuré équipé d’une forme de Dirichlet admettant un noyau de la chaleur euclidien est nécessairement isométrique à l’espace euclidien. Nous en déduisons une preuve alternative du célèbre théorème de presque rigidité du volume de Colding grâce à une version quantitative de notre résultat principal. Nous traitons aussi le cas d’un espace métrique mesuré équipé d’une forme de Dirichlet admettant un noyau de la chaleur sphérique.

We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding’s celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.

Reçu le : 2021-03-11
Accepté le : 2021-10-25
Publié le : 2021-11-17

DOI : 10.5802/jep.179

Classification : 35K08, 31C25, 53C23, 53C21
Keywords: Heat kernel, harmonic functions, asymptotic cone
Mot clés : Noyau de la chaleur, fonctions harmoniques, cône asymptotique

Affiliations des auteurs :

Carron, Gilles ¹ ; Tewodrose, David ²

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Carron, Gilles; Tewodrose, David. A rigidity result for metric measure spaces with Euclidean heat kernel. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 101-154. doi : 10.5802/jep.179. http://www.numdam.org/articles/10.5802/jep.179/

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