Convergence of Riemannian manifolds and Laplace operators. I

Atsushi Kasue[1]

  • [1] Osaka City University, Department of Mathematics, Sugimoto, Sumiyoshi, Osaka 558-8585 (Japon) et Kanazawa University, Department of Mathematics, Kanazawa 920-1192 (Japon)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 4, page 1219-1257
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Kasue, Atsushi. "Convergence of Riemannian manifolds and Laplace operators. I." Annales de l’institut Fourier 52.4 (2002): 1219-1257. <http://eudml.org/doc/116008>.

@article{Kasue2002,
abstract = {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.},
affiliation = {Osaka City University, Department of Mathematics, Sugimoto, Sumiyoshi, Osaka 558-8585 (Japon) et Kanazawa University, Department of Mathematics, Kanazawa 920-1192 (Japon)},
author = {Kasue, Atsushi},
journal = {Annales de l’institut Fourier},
keywords = {Laplace operator; energy form; heat kernel; spectral convergence; Gromov-Hausdorff distance; eigenvalue},
language = {eng},
number = {4},
pages = {1219-1257},
publisher = {Association des Annales de l'Institut Fourier},
title = {Convergence of Riemannian manifolds and Laplace operators. I},
url = {http://eudml.org/doc/116008},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Kasue, Atsushi
TI - Convergence of Riemannian manifolds and Laplace operators. I
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 4
SP - 1219
EP - 1257
AB - 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.
LA - eng
KW - Laplace operator; energy form; heat kernel; spectral convergence; Gromov-Hausdorff distance; eigenvalue
UR - http://eudml.org/doc/116008
ER -

References

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  1. K. Akutagawa, Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl 4 (1994), 239-258 Zbl0810.53030MR1299397
  2. K. Akutagawa, Convergence for Yamabe metrics of positive scalar curvature with integral bound on curvature, Pacific J. Math 175 (1996), 239-258 Zbl0881.53036MR1432834
  3. P. Bérard, G. Besson, S. Gallot, On embedding Riemannian manifolds in a Hilbert space using their heat kernels, (1988) Zbl0806.53044
  4. P. Bérard, G. Besson, S. Gallot, Embedding Riemannian manifolds by their heat kernel, Geom. Funct. Anal 4 (1994), 373-398 Zbl0806.53044MR1280119
  5. G. Besson, A Kato type inequality for Riemannian submersion with totally geodesic fibers, Ann. Glob. Analysis and Geometry 4 (1986), 273-289 Zbl0631.53035MR910547
  6. M. Biroli, U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pure Appl (4) 169 (1995), 125-181 Zbl0851.31008MR1378473
  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) 1 (1996), 205-232, Soc. Math. France Zbl0884.58088
  8. J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal 9 (1999), 428-517 Zbl0942.58018MR1708448
  9. E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math 109 (1987), 319-334 Zbl0659.35009MR882426
  10. K. Fukaya, Collapsing Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math 87 (1987), 517-547 Zbl0589.58034MR874035
  11. M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, (1994), Walter de Gruyter, Berlin-New York Zbl0838.31001MR1303354
  12. M. Gromov, Structures métriques pour les variétés riemanniennes, (1981), Cedic Fernand-Nathan, Paris Zbl0509.53034MR682063
  13. A. Grigor'yan, Heat kernel of a noncompact Riemannian manifold, Stochastic Analysis (Ithaca, NY, 1993) 57 (1993), 239-263, Amer. Math. Soc, Providence, R.I. Zbl0829.58041
  14. Y. Hashimoto, S. Manabe, Y. Ogura, Short time asymptotics and an approximation for the heat kernel of a singular diffusion, Itô's Stochastic Calculus and Probability Theory (1996), 129-140, Springer-Verlag, Tokyo Zbl0866.60066
  15. J. Heinonen, P. Koskela, Quasi conformal maps on metric spaces with controlled geometry, Acta Math 181 (1998), 1-61 Zbl0915.30018
  16. N. Ikeda, Y. Ogura, Degenerating sequences of Riemannian metrics on a manifold and their Brownian motions, Diffusions in Analysis and Geometry (1990), 293-312, Birkhäuser, Boston-Bassel-Berlin Zbl0742.58058
  17. D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J 53 (1986), 503-523 Zbl0614.35066MR850547
  18. A. Kasue, H. Kumura, Spectral convergence of Riemannian manifolds, Tohoku Math. J 46 (1994), 147-179 Zbl0814.53035MR1272877
  19. A. Kasue, H. Kumura, Spectral convergence of Riemannian manifolds, II, Tohoku Math. J 48 (1996), 71-120 Zbl0853.58100MR1373175
  20. A. Kasue, H. Kumura, Y. Ogura, Convergence of heat kernels on a compact manifold, Kyuushu J. Math 51 (1997), 453-524 Zbl0914.58031MR1470166
  21. A. Kasue, H. Kumura, Spectral convergence of conformally immersed surfaces with bounded mean curvature Zbl1043.58013MR1916863
  22. A. Kasue, Convergence of Riemannian manifolds and Laplace operators; II (in preparation) Zbl1099.53033
  23. A. Nagel, E. M. Stein, S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math 55 (1985), 103-147 Zbl0578.32044MR793239
  24. Y. Ogura, Weak convergence of laws of stochastic processes on Riemannian manifolds, Probab. Theory Relat. Fields 119 (2001), 529-557 Zbl0983.58017MR1826406
  25. J.A. Ramírez, Short-time asymptotics in Dirichlet spaces, Comm. Pure Appl. Math 54 (2001), 259-293 Zbl1023.60049MR1809739
  26. L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequality, Duke Math. J., Int. Math. Res. Notices 2 (1992), 27-38 Zbl0769.58054MR1150597
  27. K. T. Sturm, Analysis on local Dirichlet spaces I. Recurrence,conservativeness and L p -Liouville properties, J. Reine Angew. Math. 456 (1994), 173-196 Zbl0806.53041MR1301456
  28. K. T. Sturm, Analysis on local Dirichlet spaces II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math 32 (1995), 275-312 Zbl0854.35015MR1355744
  29. K. T. Sturm, Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, J. Math. Pures Appl 75 (1996), 273-297 Zbl0854.35016MR1387522
  30. K. Yoshikawa, Degeneration of algebraic manifolds and the continuity of the spectrum of the Laplacian, Nagoya Math. J 146 (1997), 83-129 Zbl0880.58030MR1460955

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