Rough isometries, and combinatorial approximations of geometries of non-compact riemannian manifolds

Masahiko KANAI

doi:10.2969/jmsj/03730391

Masahiko KANAI

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JOURNAL FREE ACCESS

1985 Volume 37 Issue 3 Pages 391-413

DOI https://doi.org/10.2969/jmsj/03730391

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Published: 1985 Received: May 01, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598.
2) A. Avez, Variétés riemanniennes sans points focaux, C. R. Acad. Sci. Paris Ser. A, 270 (1970), 188-191.
3) E. Bombieri and E. Guisti, Harnack's inequality for elliptic differential equations on minimal surfaces, Invent. Math., 15 (1972), 24-46.
4) R. Brooks, A relation between growth and the spectrum of the laplacian, Math. Z., 178 (1981), 501-508.
5) R. Brooks, The fundamental group and the spectrum of the laplacian, Comment. Math. Helv., 56 (1981), 581-598.
6) P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup., 15 (1982), 213-230.
7) S.-Y. Cheng and S.-T. Yau, Differential equations on riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28 (1975), 333-354.
8) C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup., 13 (1980), 419-435.
9) R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., 699, Springer, 1979.
10) M. Gromov, Hyperbolic manifolds, groups and actions, Ann. of Math. Studies, 97, Princeton Univ. Press, Princeton, 1981, 183-213.
11) M. Gromov, Structures métriques pour les variétés riemanniennes, Cedic, Paris, 1981.
12) A. Manning, Topological entropy for geodesic flows, Ann. of Math., 110 (1979), 567-573.
13) J. Milnor, A note on curvature and fundamental group, J. Diff. Geom., 2 (1968), 1-7.
14) J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.
15) G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies, 78, Princeton Univ. Press, Princeton, 1973.
16) R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc., 84 (1978), 1182-1238.
17) A. Phillips and D. Sullivan, Geometry of leaves, Topology, 20 (1981), 209-218.
18) S.-T. Yau, Harmonic functions on complete riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.
19) S.-T. Yau, Survey on partial differential equations in differential geometry, Ann. of Math. Studies, 102, Princeton Univ. Press, Princeton, 1982, 3-71.

Right : [1] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598.
[2] A. Avez, Variétés riemanniennes sans points focaux, C. R. Acad. Sci. Paris Ser. A, 270 (1970), 188-191.
[3] E. Bombieri and E. Guisti, Harnack's inequality for elliptic differential equations on minimal surfaces, Invent. Math., 15 (1972), 24-46.
[4] R. Brooks, A relation between growth and the spectrum of the laplacian, Math. Z., 178 (1981), 501-508.
[5] R. Brooks, The fundamental group and the spectrum of the laplacian, Comment. Math. Helv., 56 (1981), 581-598.
[6] P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup., 15 (1982), 213-230.
[7] S. -Y. Cheng and S. -T. Yau, Differential equations on riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28 (1975), 333-354.
[8] C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup., 13 (1980), 419-435.
[9] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., 699, Springer, 1979.
[10] M. Gromov, Hyperbolic manifolds, groups and actions, Ann. of Math. Studies, 97, Princeton Univ. Press, Princeton, 1981, 183-213.
[11] M. Gromov, Structures métriques pour les variétés riemanniennes, Cedic, Paris, 1981.
[12] A. Manning, Topological entropy for geodesic flows, Ann. of Math., 110 (1979), 567-573.
[13] J. Milnor, A note on curvature and fundamental group, J. Diff. Geom., 2 (1968), 1-7.
[14] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.
[15] G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies, 78, Princeton Univ. Press, Princeton, 1973.
[16] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc., 84 (1978), 1182-1238.
[17] A. Phillips and D. Sullivan, Geometry of leaves, Topology, 20 (1981), 209-218.
[18] S. -T. Yau, Harmonic functions on complete riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.
[19] S. -T. Yau, Survey on partial differential equations in differential geometry, Ann. of Math. Studies, 102, Princeton Univ. Press, Princeton, 1982, 3-71.

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