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.
Accepted:
Published online:
Keywords: Heat kernel, harmonic functions, asymptotic cone
Mots-clés : Noyau de la chaleur, fonctions harmoniques, cône asymptotique
Gilles Carron 1; David Tewodrose 2

@article{JEP_2022__9__101_0, author = {Gilles Carron and David Tewodrose}, title = {A rigidity result for metric measure spaces with {Euclidean} heat kernel}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {101--154}, publisher = {\'Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.179}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.179/} }
TY - JOUR AU - Gilles Carron AU - David Tewodrose TI - A rigidity result for metric measure spaces with Euclidean heat kernel JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 101 EP - 154 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.179/ DO - 10.5802/jep.179 LA - en ID - JEP_2022__9__101_0 ER -
%0 Journal Article %A Gilles Carron %A David Tewodrose %T A rigidity result for metric measure spaces with Euclidean heat kernel %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 101-154 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.179/ %R 10.5802/jep.179 %G en %F JEP_2022__9__101_0
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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