A rigidity result for metric measure spaces with Euclidean heat kernel

Gilles Carron; David Tewodrose

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]

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

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 :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A rigidity result for metric measure spaces with {Euclidean} heat kernel},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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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, Tome 9 (2022), pp. 101-154. doi : 10.5802/jep.179. https://jep.centre-mersenne.org/articles/10.5802/jep.179/

Bibliographie
Cité par

[ACDM15] L. Ambrosio, M. Colombo & S. Di Marino - “Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope”, in Variational methods for evolving objects, Adv. Stud. Pure Math., vol. 67, Mathematical Society of Japan, Tokyo, 2015, p. 1-58 | DOI | MR | Zbl

[ACT21] C. L. Aldana, G. Carron & S. Tapie - “ $A_{\infty}$ weights and compactness of conformal metrics under $L^{n / 2}$ curvature bounds”, Anal. PDE 14 (2021) no. 7, p. 2163-2205 | DOI

[AFP00] L. Ambrosio, N. Fusco & D. Pallara - Functions of bounded variation and free discontinuity problems, Oxford Math. Monographs, The Clarendon Press, Oxford University Press, New York, 2000 | Zbl

[AGG19] T. Adamowicz, M. Gaczkowski & P. Górka - “Harmonic functions on metric measure spaces”, Rev. Mat. Univ. Complut. Madrid 32 (2019) no. 1, p. 141-186 | DOI | MR | Zbl

[AH12] K. Atkinson & W. Han - Spherical harmonics and approximations on the unit sphere: an introduction, Lect. Notes in Math., vol. 2044, Springer, Heidelberg, 2012 | DOI | MR | Zbl

[AT04] L. Ambrosio & P. Tilli - Topics on analysis in metric spaces, Oxford Lecture Series in Math. and its Applications, vol. 25, Oxford University Press, Oxford, 2004 | MR | Zbl

[BBI01] D. Burago, Y. Burago & S. Ivanov - A course in metric geometry, Graduate Studies in Math., vol. 33, American Mathematical Society, Providence, RI, 2001 | DOI | MR

[BD59] A. Beurling & J. Deny - “Dirichlet spaces”, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), p. 208-215 | DOI | Zbl

[Car19] G. Carron - “Geometric inequalities for manifolds with Ricci curvature in the Kato class”, Ann. Inst. Fourier (Grenoble) 69 (2019) no. 7, p. 3095-3167 | DOI | Numdam | MR | Zbl

[CC97] J. Cheeger & T. H. Colding - “On the structure of spaces with Ricci curvature bounded below. I”, J. Differential Geom. 46 (1997) no. 3, p. 406-480 | MR | Zbl

[Che99] J. Cheeger - “Differentiability of Lipschitz functions on metric measure spaces”, Geom. Funct. Anal. 9 (1999) no. 3, p. 428-517 | DOI | MR | Zbl

[CM97] T. H. Colding & W. P. Minicozzi - “Harmonic functions on manifolds”, Ann. of Math. (2) 146 (1997) no. 3, p. 725-747 | DOI | MR | Zbl

[Coh07] A. M. Cohen - Numerical methods for Laplace transform inversion, Numerical Methods and Algorithms, vol. 5, Springer, New York, 2007 | MR | Zbl

[Col97] T. H. Colding - “Ricci curvature and volume convergence”, Ann. of Math. (2) 145 (1997) no. 3, p. 477-501 | DOI | MR | Zbl

[CY81] J. Cheeger & S. T. Yau - “A lower bound for the heat kernel”, Comm. Pure Appl. Math. 34 (1981) no. 4, p. 465-480 | DOI | MR | Zbl

[FOT11] M. Fukushima, Y. Oshima & M. Takeda - Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Math., vol. 19, Walter de Gruyter & Co., Berlin, 2011 | MR | Zbl

[GG09] M. Gaczkowski & P. Górka - “Harmonic functions on metric measure spaces: convergence and compactness”, Potential Anal. 31 (2009) no. 3, p. 203-214 | DOI | MR | Zbl

[Gri91] A. A. Grigor’yan - “The heat equation on noncompact Riemannian manifolds”, Mat. Sb. 182 (1991) no. 1, p. 55-87 | Zbl

[Gri94] A. A. Grigor’yan - “Integral maximum principle and its applications”, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) no. 2, p. 353-362 | DOI | MR | Zbl

[Gri10] A. A. Grigor’yan - “Heat kernels on metric measure spaces with regular volume growth”, in Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM), vol. 13, Int. Press, Somerville, MA, 2010, p. 1-60 | MR | Zbl

[Gro07] M. Gromov - Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007 | Zbl

[Hei01] J. Heinonen - Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001 | DOI | Zbl

[HKX16] B. Hua, M. Kell & C. Xia - “Harmonic functions on metric measure spaces”, 2016 | arXiv

[Hon15] S. Honda - “Ricci curvature and $L^{p}$ -convergence”, J. reine angew. Math. 705 (2015), p. 85-154 | DOI | MR | Zbl

[Hua11] B. Hua - “Harmonic functions of polynomial growth on singular spaces with nonnegative Ricci curvature”, Proc. Amer. Math. Soc. 139 (2011) no. 6, p. 2191-2205 | DOI | MR | Zbl

[KZ12] P. Koskela & Y. Zhou - “Geometry and analysis of Dirichlet forms”, Adv. Math. 231 (2012) no. 5, p. 2755-2801 | DOI | MR | Zbl

[LTW97] P. Li, L.-F. Tam & J. Wang - “Sharp bounds for the Green’s function and the heat kernel”, Math. Res. Lett. 4 (1997) no. 4, p. 589-602 | DOI | MR | Zbl

[LY86] P. Li & S.-T. Yau - “On the parabolic kernel of the Schrödinger operator”, Acta Math. 156 (1986) no. 3-4, p. 153-201 | DOI

[MP49] S. Minakshisundaram & A. Pleijel - “Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds”, Canad. J. Math. 1 (1949), p. 242-256 | DOI | MR | Zbl

[RS80] M. Reed & B. Simon - Methods of modern mathematical physics I. Functional analysis, Academic Press, Inc., New York, 1980 | Zbl

[SC92] L. Saloff-Coste - “A note on Poincaré, Sobolev, and Harnack inequalities”, Internat. Math. Res. Notices (1992) no. 2, p. 27-38 | DOI | Zbl

[Stu94] K.-T. Sturm - “Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^{p}$ -Liouville properties”, J. reine angew. Math. 456 (1994), p. 173-196 | DOI | MR | Zbl

[Stu95] K.-T. Sturm - “Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations”, Osaka J. Math. 32 (1995) no. 2, p. 275-312 | MR | Zbl

[Stu96] K.-T. Sturm - “Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality”, J. Math. Pures Appl. (9) 75 (1996) no. 3, p. 273-297 | MR | Zbl

[tERS07] A. F. M. ter Elst, D. W. Robinson & A. Sikora - “Small time asymptotics of diffusion processes”, J. Evol. Equ. 7 (2007) no. 1, p. 79-112 | DOI | MR | Zbl

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