A rigidity result for metric measure spaces with Euclidean heat kernel
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 101-154.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.179
Classification: 35K08,  31C25,  53C23,  53C21
Keywords: Heat kernel, harmonic functions, asymptotic cone
Gilles Carron 1; David Tewodrose 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Gilles Carron; David Tewodrose. A rigidity result for metric measure spaces with Euclidean heat kernel. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 101-154. doi : 10.5802/jep.179. https://jep.centre-mersenne.org/articles/10.5802/jep.179/

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