메뉴 건너뛰기




Volumn 11, Issue 4, 2003, Pages 599-673

Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry

Author keywords

[No Author keywords available]

Indexed keywords


EID: 4043070671     PISSN: 10198385     EISSN: None     Source Type: Journal    
DOI: 10.4310/CAG.2003.v11.n4.a1     Document Type: Article
Times cited : (165)

References (59)
  • 1
    • 38249039168 scopus 로고
    • Convergence of Dirichlet forms and associated Schrödinger operators
    • S. Albeverio, S. Kusuoka, and L. Streit, Convergence of Dirichlet forms and associated Schrödinger operators, J. Funct. Anal. 68 (1986), no. 2, 130-148.
    • (1986) J. Funct. Anal. , vol.68 , Issue.2 , pp. 130-148
    • Albeverio, S.1    Kusuoka, S.2    Streit, L.3
  • 2
    • 85039506855 scopus 로고
    • Fonctions propres sur des variétés avec des anses fines, application à la multiplicité
    • Année 1988-1989, Univ. Grenoble I, Saint
    • C. Anné, Fonctions propres sur des variétés avec des anses fines, application à la multiplicité, Séminaire de Théorie Spectrale et Géométrie, No. 7, Année 1988-1989, Univ. Grenoble I, Saint, 1989, pp. 123-133.
    • (1989) Séminaire de Théorie Spectrale et Géométrie , Issue.7 , pp. 123-133
    • Anné, C.1
  • 3
    • 85039507959 scopus 로고    scopus 로고
    • Formes différentielles sur des variétés avec des anses fines
    • (Luminy, 1992), Soc. Math. France, Paris
    • _, Formes différentielles sur des variétés avec des anses fines, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Soc. Math. France, Paris, 1996, pp. 69-76.
    • (1996) Actes de la Table Ronde de Géométrie Différentielle , pp. 69-76
  • 4
    • 0007280787 scopus 로고
    • Opérateur de Hodge-Laplace sur des variétés compactes privées d'un nombre fini de boules
    • C. Anne and B. Colbois, Opérateur de Hodge-Laplace sur des variétés compactes privées d'un nombre fini de boules, J. Funct. Anal. 115 (1993), no. 1, 190-211.
    • (1993) J. Funct. Anal. , vol.115 , Issue.1 , pp. 190-211
    • Anne, C.1    Colbois, B.2
  • 5
    • 0007208489 scopus 로고
    • Spectre du laplacien agissant sur les p-formes différentielles et écrasement d'anses
    • _, Spectre du laplacien agissant sur les p-formes différentielles et écrasement d'anses, Math. Ann. 303 (1995), no. 3, 545-573.
    • (1995) Math. Ann. , vol.303 , Issue.3 , pp. 545-573
  • 7
    • 0001199139 scopus 로고
    • Embedding riemannian manifolds by their heat kernel
    • P. Bérard, G. Besson, and S. Gallot, Embedding Riemannian manifolds by their heat kernel, Geom. Funct. Anal. 4 (1994), no. 4, 373-398.
    • (1994) Geom. Funct. Anal. , vol.4 , Issue.4 , pp. 373-398
    • Bérard, P.1    Besson, G.2    Gallot, S.3
  • 8
    • 0001787207 scopus 로고
    • A. D. Aleksandrov spaces with curvatures bounded below
    • Yu. Burago, M. Gromov, and G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3-51, 222, translation in Russian Math. Surveys 47 (1992), no. 2, 1-58.
    • (1992) Uspekhi Mat. Nauk , vol.47 , Issue.2-284 , pp. 3-51
    • Burago, Yu.1    Gromov, M.2    Perel'man, G.3
  • 9
    • 4043098146 scopus 로고
    • Yu. Burago, M. Gromov, and G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3-51, 222, translation in Russian Math. Surveys 47 (1992), no. 2, 1-58.
    • (1992) Translation in Russian Math. Surveys , vol.47 , Issue.2 , pp. 1-58
  • 10
    • 0000720752 scopus 로고
    • A note on the isoperimetric constant
    • P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213-230.
    • (1982) Ann. Sci. École Norm. Sup. (4) , vol.15 , Issue.2 , pp. 213-230
    • Buser, P.1
  • 11
    • 0000710192 scopus 로고
    • Spectra of domains in compact manifolds
    • I. Chavel and E. A. Feldman, Spectra of domains in compact manifolds, J. Funct. Anal. 30 (1978), no. 2, 198-222.
    • (1978) J. Funct. Anal. , vol.30 , Issue.2 , pp. 198-222
    • Chavel, I.1    Feldman, E.A.2
  • 12
    • 51249185116 scopus 로고
    • Spectra of manifolds with small handles
    • _, Spectra of manifolds with small handles, Comment. Math. Helv. 56 (1981), no. 1, 83-102.
    • (1981) Comment. Math. Helv. , vol.56 , Issue.1 , pp. 83-102
  • 13
    • 4043069693 scopus 로고
    • Isoperimetric constants of manifolds with small handles
    • _, Isoperimetric constants of manifolds with small handles, Math. Z. 184 (1983), no. 4, 435-448.
    • (1983) Math. Z. , vol.184 , Issue.4 , pp. 435-448
  • 14
    • 4043122360 scopus 로고
    • Diffusion on manifolds with small handles
    • _, Diffusion on manifolds with small handles, Indiana Univ. Math. J. 34 (1985), no. 3, 449-461.
    • (1985) Indiana Univ. Math. J. , vol.34 , Issue.3 , pp. 449-461
  • 15
    • 20344362744 scopus 로고
    • A lower bound for the smallest eigenvalue of the Laplacian
    • (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, N. J
    • J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, N. J., 1970, pp. 195-199.
    • (1970) Problems in Analysis , pp. 195-199
    • Cheeger, J.1
  • 16
    • 0033456306 scopus 로고    scopus 로고
    • Differentiability of Lipschitz functions on metric measure spaces
    • _, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428-517.
    • (1999) Geom. Funct. Anal. , vol.9 , Issue.3 , pp. 428-517
  • 17
    • 0002157976 scopus 로고    scopus 로고
    • On the structure of spaces with Ricci curvature bounded below. III
    • J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), no. 1, 37-74.
    • (2000) J. Differential Geom. , vol.54 , Issue.1 , pp. 37-74
    • Cheeger, J.1    Colding, T.H.2
  • 18
    • 0040847204 scopus 로고
    • Convergence de variétés et convergence du spectre du Laplacien
    • B. Colbois and G. Courtois, Convergence de variétés et convergence du spectre du Laplacien, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 4, 507-518.
    • (1991) Ann. Sci. École Norm. Sup. (4) , vol.24 , Issue.4 , pp. 507-518
    • Colbois, B.1    Courtois, G.2
  • 21
    • 4043146310 scopus 로고
    • Intrinsic distance on a Lipschitz Riemannian manifold
    • G. De Cecco and G. Palmieri, Intrinsic distance on a Lipschitz Riemannian manifold, Rend. Sem. Mat. Univ. Politec. Torino 46 (1988), no. 2, 157-170 (1990).
    • (1988) Rend. Sem. Mat. Univ. Politec. Torino , vol.46 , Issue.2 , pp. 157-170
    • De Cecco, G.1    Palmieri, G.2
  • 22
    • 0003346795 scopus 로고
    • Measure theory and fine properties of functions
    • CRC Press, Boca Raton, FL
    • L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
    • (1992) Studies in Advanced Mathematics
    • Evans, L.C.1    Gariepy, R.F.2
  • 23
    • 0011194265 scopus 로고
    • Convergence of the eigenvalues of Laplacians in a class of finite graphs
    • K. Fujiwara, Convergence of the eigenvalues of Laplacians in a class of finite graphs, Contemp. Math. 173 (1994), 115-120.
    • (1994) Contemp. Math. , vol.173 , pp. 115-120
    • Fujiwara, K.1
  • 24
    • 0040460272 scopus 로고
    • Collapsing of Riemannian manifolds and eigenvalues of Laplace operator
    • K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), 517-547.
    • (1987) Invent. Math. , vol.87 , pp. 517-547
    • Fukaya, K.1
  • 25
    • 0003228611 scopus 로고
    • Dirichlet forms and symmetric Markov processes
    • Walter de Gruyter, Berlin-New York
    • M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Math., vol. 19, Walter de Gruyter, Berlin-New York, 1994.
    • (1994) De Gruyter Studies in Math. , vol.19
    • Fukushima, M.1    Oshima, Y.2    Takeda, M.3
  • 26
    • 0002253552 scopus 로고
    • The harmonic operator for exterior differential forms
    • M. P. Gaffney, The harmonic operator for exterior differential forms, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 48-50.
    • (1951) Proc. Nat. Acad. Sci. U. S. A. , vol.37 , pp. 48-50
    • Gaffney, M.P.1
  • 27
    • 0001468554 scopus 로고
    • A special stokes's theorem for complete riemannian manifolds
    • _, A special Stokes's theorem for complete Riemannian manifolds, Ann. of Math. (2) 60 (1954), 140-145.
    • (1954) Ann. of Math. (2) , vol.60 , pp. 140-145
  • 29
    • 0004076870 scopus 로고    scopus 로고
    • Birkhäuser Boston Inc., Boston, MA, Based on the 1981 French original [MR 85e:53051], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates
    • M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser Boston Inc., Boston, MA, 1999, Based on the 1981 French original [MR 85e:53051], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.
    • (1999) Metric Structures for Riemannian and Non-Riemannian Spaces
    • Gromov, M.1
  • 30
    • 0011366069 scopus 로고
    • A degenerating sequence of Riemannian metrics on a manifold and their Brownian motions
    • Evanston, IL, Birkhäuser Boston, Boston, MA, 1990
    • N. Ikeda and Y. Ogura, A degenerating sequence of Riemannian metrics on a manifold and their Brownian motions, Diffusion processes and related problems in analysis, Vol. I (Evanston, IL, 1989), Birkhäuser Boston, Boston, MA, 1990, pp. 293-312.
    • (1989) Diffusion Processes and Related Problems in Analysis , vol.1 , pp. 293-312
    • Ikeda, N.1    Ogura, Y.2
  • 33
    • 84972563105 scopus 로고
    • Spectral convergence of Riemannian manifolds
    • A. Kasue and H. Kumura, Spectral convergence of Riemannian manifolds, Tôhoku Math. J. (2) 48 (1994), no. 2, 147-179.
    • (1994) Tôhoku Math. J. (2) , vol.48 , Issue.2 , pp. 147-179
    • Kasue, A.1    Kumura, H.2
  • 34
    • 0030304680 scopus 로고    scopus 로고
    • Spectral convergence of Riemannian manifolds. II
    • _, Spectral convergence of Riemannian manifolds. II, Tôhoku Math. J. (2) 48 (1996), no. 1, 71-120.
    • (1996) Tôhoku Math. J. (2) , vol.48 , Issue.1 , pp. 71-120
  • 35
    • 84867971784 scopus 로고    scopus 로고
    • Spectral convergence of conformally immersed surfaces with bounded mean curvature
    • _, Spectral convergence of conformally immersed surfaces with bounded mean curvature, J. Geom. Anal. 12 (2002), no. 4, 663-681.
    • (2002) J. Geom. Anal. , vol.12 , Issue.4 , pp. 663-681
  • 36
    • 0009324274 scopus 로고    scopus 로고
    • Convergence of heat kernels on a compact manifold
    • A. Kasue, H. Kumura, and Y. Ogura, Convergence of heat kernels on a compact manifold, Kyushu J. Math. 51 (1997), no. 2, 453-524.
    • (1997) Kyushu J. Math. , vol.51 , Issue.2 , pp. 453-524
    • Kasue, A.1    Kumura, H.2    Ogura, Y.3
  • 37
    • 0003862232 scopus 로고
    • Springer- Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften
    • T. Kato, Perturbation theory for linear operators, second ed., Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, Band 132.
    • (1976) Perturbation Theory for Linear Operators, Second Ed. , vol.132
    • Kato, T.1
  • 38
    • 0039675398 scopus 로고    scopus 로고
    • Beginning of analysis on Alexandrov spaces
    • Amer. Math. Soc., Providence, RI
    • K. Kuwae, Y. Machigashira, and T. Shioya, Beginning of analysis on Alexandrov spaces, Geometry and topology: Aarhus (1998), Amer. Math. Soc., Providence, RI, 2000, pp. 275-284.
    • (1998) Geometry and Topology: Aarhus , pp. 275-284
    • Kuwae, K.1    Machigashira, Y.2    Shioya, T.3
  • 39
    • 0035639753 scopus 로고    scopus 로고
    • Sobolev spaces, Laplacian, and heat kernel on alexandrov spaces
    • _, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269-316.
    • (2001) Math. Z. , vol.238 , Issue.2 , pp. 269-316
  • 41
    • 0002630730 scopus 로고
    • Estimates of eigenvalues of a compact Riemannian manifold
    • (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1997), Amer. Math. Soc., Providence, R.I.
    • P. Li and S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Amer. Math. Soc., Providence, R.I., 1980, pp. 205-239.
    • (1980) Geometry of the Laplace Operator , pp. 205-239
    • Li, P.1    Yau, S.T.2
  • 42
    • 0033475623 scopus 로고    scopus 로고
    • Essential self-adjointness of Laplacians on Riemannian manifolds with fractal boundary
    • J. Masamune, Essential self-adjointness of Laplacians on Riemannian manifolds with fractal boundary, Comm. Partial Differential Equations 24 (1999), no. 3-4, 749-757.
    • (1999) Comm. Partial Differential Equations , vol.24 , Issue.3-4 , pp. 749-757
    • Masamune, J.1
  • 43
    • 0000548556 scopus 로고
    • Composite media and asymptotic Dirichlet forms
    • U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994), no. 2, 368-421.
    • (1994) J. Funct. Anal. , vol.123 , Issue.2 , pp. 368-421
    • Mosco, U.1
  • 44
    • 0002351322 scopus 로고    scopus 로고
    • Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds
    • J. R. Norris, Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds, Acta Math. 179 (1997), 79-103.
    • (1997) Acta Math. , vol.179 , pp. 79-103
    • Norris, J.R.1
  • 46
    • 84972500357 scopus 로고
    • The Riemannian structure of Alexandrov spaces
    • Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994), no. 3, 629-658.
    • (1994) J. Differential Geom. , vol.39 , Issue.3 , pp. 629-658
    • Otsu, Y.1    Shioya, T.2
  • 49
    • 0040106169 scopus 로고
    • enlarged ed., Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Band 228
    • I. E. Segal and R. A. Kunze, Integrals and operators, enlarged ed., Springer-Verlag, Berlin, 1978, Grundlehren der Mathematischen Wissenschaften, Band 228.
    • (1978) Integrals and Operators
    • Segal, I.E.1    Kunze, R.A.2
  • 50
    • 51249162132 scopus 로고
    • Mass of rays in Alexandrov spaces of nonnegative curvature
    • T. Shioya, Mass of rays in Alexandrov spaces of nonnegative curvature, Comment. Math. Helv. 69 (1994), no. 2, 208-228.
    • (1994) Comment. Math. Helv. , vol.69 , Issue.2 , pp. 208-228
    • Shioya, T.1
  • 51
    • 22644448306 scopus 로고    scopus 로고
    • The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature
    • _, The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature, Trans. Amer. Math. Soc. 351 (1999), no. 5, 1765-1801.
    • (1999) Trans. Amer. Math. Soc , vol.351 , Issue.5 , pp. 1765-1801
  • 52
    • 85037882492 scopus 로고    scopus 로고
    • Convergence of Alexandrov spaces and spectrum of Laplacian
    • _, Convergence of Alexandrov spaces and spectrum of Laplacian, J. Math. Soc. Japan 53 (2001), no. 1, 1-15.
    • (2001) J. Math. Soc. Japan , vol.53 , Issue.1 , pp. 1-15
  • 54
    • 0001604599 scopus 로고
    • On the geometry defined by Dirichlet forms
    • Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Birkhäuser, Basel
    • _, On the geometry defined by Dirichlet forms, Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Progr. Probab., vol. 36, Birkhäuser, Basel, 1995, pp. 231-242.
    • (1995) Progr. Probab. , vol.36 , pp. 231-242
  • 55
    • 84991663826 scopus 로고    scopus 로고
    • Collapsing of connected sums and the eigenvalues of the Laplacian
    • J. Takahashi, Collapsing of connected sums and the eigenvalues of the Laplacian, J. Geom. Phys. 40 (2002), no. 3-4, 201-208.
    • (2002) J. Geom. Phys. , vol.40 , Issue.3-4 , pp. 201-208
    • Takahashi, J.1
  • 56
    • 0001413643 scopus 로고
    • The index of signature operators on Lipschitz manifolds
    • N. Teleman, The index of signature operators on Lipschitz manifolds, Inst. Hautes Études Sci. Publ. Math. (1983), no. 58, 39-78 (1984).
    • (1983) Inst. Hautes Études Sci. Publ. Math. , Issue.58 , pp. 39-78
    • Teleman, N.1
  • 57
    • 0013491135 scopus 로고
    • Spectral limits for hyperbolic surfaces. I, II
    • S. A. Wolpert, Spectral limits for hyperbolic surfaces. I, II, Invent. Math. 108 (1992), no. 1, 67-89, 91-129.
    • (1992) Invent. Math. , vol.108 , Issue.1 , pp. 67-89
    • Wolpert, S.A.1
  • 58
    • 0000100131 scopus 로고
    • Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold
    • S. T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487-507.
    • (1975) Ann. Sci. École Norm. Sup. (4) , vol.8 , Issue.4 , pp. 487-507
    • Yau, S.T.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.