Rough isometries and the parabolicity of riemannian manifolds

Masahiko KANAI

doi:10.2969/jmsj/03820227

Masahiko KANAI

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JOURNAL FREE ACCESS

1986 Volume 38 Issue 2 Pages 227-238

DOI https://doi.org/10.2969/jmsj/03820227

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Published: 1986 Received: October 20, 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) P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup., 15 (1982), 213-230.
2) C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup., 13(1980), 419-435.
3) J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., 32 (1983), 703-716.
4) J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., 284 (1984), 787-794.
5) E. B. Dynkin and A. H. Yushkevich, Markov Processes; Theorems and Problems, Plenum Press, New York, 1969.
6) D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math., 33(1980), 199-211.
7) D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer, New York, 1983.
8) S. Ito, Fundamental solutions of parabolic differential equations and boundary value problems, Japan. J. Math., 27 (1957), 55-102.
9) S. Ito, On existence of Green function and positive superharmonic functions for linear elliptic operators of second order, J. Math. Soc. Japan, 16(1964), 299-306.
10) M. Kanai, Rough isometries, and combinatorial approximations of geometries of non-compact riemannian manifolds, J. Math. Soc. Japan, 37 (1985), 391-413.
11) A. Kasue, A laplacian comparison theorem and function theoretic properties of a complete riemannian manifold, Japan. J. Math., 8 (1982), 309-341.
12) T. Lyons and D. Sullivan, Function theory, random paths, and covering spaces, J. Diff. Geom., 19 (1984), 299-323.
13) N. Th. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier (Grenoble), 34 (1984), 243-269.

Right : [1] P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup., 15 (1982), 213-230.
[2] C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup., 13(1980), 419-435.
[3] J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., 32 (1983), 703-716.
[4] J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., 284 (1984), 787-794.
[5] E. B. Dynkin and A. H. Yushkevich, Markov Processes; Theorems and Problems, Plenum Press, New York, 1969.
[6] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math., 33 (1980), 199-211.
[7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer, New York, 1983.
[8] S. Itô, Fundamental solutions of parabolic differential equations and boundary value problems, Japan. J. Math., 27 (1957), 55-102.
[9] S. Itô, On existence of Green function and positive superharmonic functions for linear elliptic operators of second order, J. Math. Soc. Japan, 16 (1964), 299-306.
[10] M. Kanai, Rough isometries, and combinatorial approximations of geometries of non-compact riemannian manifolds, J. Math. Soc. Japan, 37 (1985), 391-413.
[11] A. Kasue, A laplacian comparison theorem and function theoretic properties of a complete riemannian manifold, Japan. J. Math., 8 (1982), 309-341.
[12] T. Lyons and D. Sullivan, Function theory, random paths, and covering spaces, J. Diff. Geom., 19 (1984), 299-323.
[13] N. Th. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier (Grenoble), 34 (1984), 243-269.

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