Convergence of Riemannian manifolds and Laplace operators. I

Kasue, Atsushi

Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1219-1257.

Nous étudions la convergence spectrale des variétés riemanniennes compactes par rapport à la distance de Gromov-Hausdorff et discutons des distances géodésiques et des formes d'énergie des espaces de limites.

We study the spectral convergence of compact Riemannian manifolds in relation with the Gromov-Hausdorff distance and discuss the geodesic distances and the energy forms of the limit spaces.

Classification : 53C21, 58D17, 58J50
Keywords: Laplace operator, energy form, heat kernel, spectral convergence, Gromov-Hausdorff distance
Mot clés : opérateur de Laplace, forme d'énergie, noyau de la chaleur, convergence spectrale, distance de Gromov-Hausdorff

[1] K. Akutagawa Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl, Volume 4 (1994), pp. 239-258 | DOI | MR | Zbl

[2] K. Akutagawa Convergence for Yamabe metrics of positive scalar curvature with integral bound on curvature, Pacific J. Math, Volume 175 (1996), pp. 239-258 | MR | Zbl

[3] P. Bérard; G. Besson; S. Gallot On embedding Riemannian manifolds in a Hilbert space using their heat kernels (1988) (Prépublication de I'Institut Fourier, No 109)

[4] P. Bérard; G. Besson; S. Gallot Embedding Riemannian manifolds by their heat kernel, Geom. Funct. Anal, Volume 4 (1994), pp. 373-398 | DOI | MR | Zbl

[5] G. Besson A Kato type inequality for Riemannian submersion with totally geodesic fibers, Ann. Glob. Analysis and Geometry, Volume 4 (1986), pp. 273-289 | DOI | MR | Zbl

[6] M. Biroli; U. Mosco A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pure Appl (4), Volume 169 (1995), pp. 125-181 | DOI | MR | Zbl

[7] G. Carron Inégalités isopérimétriques de Faber-Krahn et conséquences, Actes de la Table Ronde de Géométrie Différentielle en l'Honneur de M. Berger (Luminy, 1992) (Sémin. Congr.), Volume 1 (1996), pp. 205-232 | Zbl

[8] J. Cheeger Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal, Volume 9 (1999), pp. 428-517 | DOI | MR | Zbl

[9] E. B. Davies Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math, Volume 109 (1987), pp. 319-334 | DOI | MR | Zbl

[10] K. Fukaya Collapsing Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math, Volume 87 (1987), pp. 517-547 | DOI | MR | Zbl

[11] M. Fukushima; Y. Oshima; M. Takeda Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin-New York, 1994 | MR | Zbl

[12] M. Gromov Structures métriques pour les variétés riemanniennes, Cedic Fernand-Nathan, Paris, 1981 | MR | Zbl

[13] A. Grigor'yan Heat kernel of a noncompact Riemannian manifold, Stochastic Analysis (Ithaca, NY, 1993) (Proc. Symposia in Pure Math), Volume 57 (1993), pp. 239-263 | Zbl

[14] Y. Hashimoto; S. Manabe; Y. Ogura; ed. N. Ikeda et al. Short time asymptotics and an approximation for the heat kernel of a singular diffusion, Itô's Stochastic Calculus and Probability Theory (1996), pp. 129-140 | Zbl

[15] J. Heinonen; P. Koskela Quasi conformal maps on metric spaces with controlled geometry, Acta Math, Volume 181 (1998), pp. 1-61 | DOI | Zbl

[16] N. Ikeda; Y. Ogura; ed. M. Pinsky Degenerating sequences of Riemannian metrics on a manifold and their Brownian motions, Diffusions in Analysis and Geometry (1990), pp. 293-312 | Zbl

[17] D. Jerison The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J, Volume 53 (1986), pp. 503-523 | MR | Zbl

[18] A. Kasue; H. Kumura Spectral convergence of Riemannian manifolds, Tohoku Math. J, Volume 46 (1994), pp. 147-179 | DOI | MR | Zbl

[19] A. Kasue; H. Kumura Spectral convergence of Riemannian manifolds, II, Tohoku Math. J, Volume 48 (1996), pp. 71-120 | DOI | MR | Zbl

[20] A. Kasue; H. Kumura; Y. Ogura Convergence of heat kernels on a compact manifold, Kyuushu J. Math, Volume 51 (1997), pp. 453-524 | DOI | MR | Zbl

[21] A. Kasue; H. Kumura Spectral convergence of conformally immersed surfaces with bounded mean curvature (To appear in J. Geom. Anal.) | MR | Zbl

[22] A. Kasue Convergence of Riemannian manifolds and Laplace operators; II (in preparation) | Zbl

[23] A. Nagel; E. M. Stein; S. Wainger Balls and metrics defined by vector fields I: Basic properties, Acta Math, Volume 55 (1985), pp. 103-147 | DOI | MR | Zbl

[24] Y. Ogura Weak convergence of laws of stochastic processes on Riemannian manifolds, Probab. Theory Relat. Fields, Volume 119 (2001), pp. 529-557 | DOI | MR | Zbl

[25] J.A. Ramírez Short-time asymptotics in Dirichlet spaces, Comm. Pure Appl. Math, Volume 54 (2001), pp. 259-293 | DOI | MR | Zbl

[26] L. Saloff-Coste A note on Poincaré, Sobolev and Harnack inequality, Duke Math. J., Int. Math. Res. Notices, Volume 2 (1992), pp. 27-38 | DOI | MR | Zbl

[27] K. T. Sturm Analysis on local Dirichlet spaces I. Recurrence,conservativeness and $L^p$-Liouville properties, J. Reine Angew. Math., Volume 456 (1994), pp. 173-196 | DOI | MR | Zbl

[28] K. T. Sturm Analysis on local Dirichlet spaces II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math, Volume 32 (1995), pp. 275-312 | MR | Zbl

[29] K. T. Sturm Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, J. Math. Pures Appl, Volume 75 (1996), pp. 273-297 | MR | Zbl

[30] K. Yoshikawa Degeneration of algebraic manifolds and the continuity of the spectrum of the Laplacian, Nagoya Math. J, Volume 146 (1997), pp. 83-129 | MR | Zbl