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Volumn 126, Issue 3, 1996, Pages 565-587

Harmonic functions on planar and almost planar graphs and manifolds, via circle packings

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EID: 0030515011     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002220050109     Document Type: Article
Times cited : (73)

References (32)
  • 1
    • 0005518536 scopus 로고
    • Positive harmonic functions and hyperbolicity, potential theory, surveys and problems
    • J. Král et al., (eds.) Heidelberg: Springer-Verlag
    • Ancona, A.: Positive harmonic functions and hyperbolicity, Potential theory, surveys and problems. In: J. Král et al., (eds.) Lecture Notes in Mathematics, vol. 1344, Heidelberg: Springer-Verlag, pp. 1-23 (1988)
    • (1988) Lecture Notes in Mathematics , vol.1344 , pp. 1-23
    • Ancona, A.1
  • 2
    • 0030099453 scopus 로고    scopus 로고
    • Examples of simply-connected Liouville manifolds with positive spectrum
    • Benjamini, I., Cao, J.: Examples of simply-connected Liouville manifolds with positive spectrum, J. Diff. Geom. Appl. 6, 31-50 (1996)
    • (1996) J. Diff. Geom. Appl. , vol.6 , pp. 31-50
    • Benjamini, I.1    Cao, J.2
  • 3
    • 0030359763 scopus 로고    scopus 로고
    • Random walks and harmonic functions on infinite planar graphs, using square tilings
    • to appear
    • Benjamini, I., Schramm, O.: Random walks and harmonic functions on infinite planar graphs, using square tilings. Annals of Probability (to appear)
    • Annals of Probability
    • Benjamini, I.1    Schramm, O.2
  • 5
    • 0039807224 scopus 로고
    • Infinite graphs and nonconstant Dirichlet finite harmonic functions
    • Cartwright, D.I., Woess, W.: Infinite graphs and nonconstant Dirichlet finite harmonic functions. SIAM J. Discrete Math., 5, 380-385 (1992)
    • (1992) SIAM J. Discrete Math. , vol.5 , pp. 380-385
    • Cartwright, D.I.1    Woess, W.2
  • 6
    • 0000162896 scopus 로고
    • Un principe variationnel pour les empilements de cercles
    • Colin De Verdiére, Y.: Un principe variationnel pour les empilements de cercles. Invent. math. 104, 655-669 (1991)
    • (1991) Invent. Math. , vol.104 , pp. 655-669
    • Colin De Verdiére, Y.1
  • 7
    • 0012962459 scopus 로고
    • 2-cohomology of noncompact surfaces
    • 2-cohomology of noncompact surfaces. Tran. Amer. Math. Soc., 284, 543-565 (1984)
    • (1984) Tran. Amer. Math. Soc. , vol.284 , pp. 543-565
    • DeBaun, D.R.1
  • 8
    • 0003252136 scopus 로고
    • Random walks and electric networks
    • Math. Association of America
    • Doyle, P.G., Snell, J.L.: Random walks and electric networks. In: The Carus Math. Monographs 22, Math. Association of America, (1984)
    • (1984) The Carus Math. Monographs , vol.22
    • Doyle, P.G.1    Snell, J.L.2
  • 10
    • 0040993898 scopus 로고
    • Limits of Dirichlet finite functions along curves
    • Glasner, M., Katz, R.: Limits of Dirichlet Finite Functions Along Curves. Rocky Mountain J. Math. 12, 429-435 (1982)
    • (1982) Rocky Mountain J. Math. , vol.12 , pp. 429-435
    • Glasner, M.1    Katz, R.2
  • 11
    • 51649160427 scopus 로고
    • Groups of polynomial growth and expanding maps
    • Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. I. H. E. S. 53, 53-73 (1981)
    • (1981) Publ. Math. I. H. E. S. , vol.53 , pp. 53-73
    • Gromov, M.1
  • 12
    • 0009305212 scopus 로고
    • Hyperbolic and parabolic packings
    • He Z.-X., Schramm, O.: Hyperbolic and parabolic packings. Discrete Comput. Geom. 14, 123-149 (1995)
    • (1995) Discrete Comput. Geom. , vol.14 , pp. 123-149
    • He, Z.-X.1    Schramm, O.2
  • 14
    • 84972557750 scopus 로고
    • Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds
    • Kanai, M.: Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds. J. Math. Soc. Japan 37, 391-413 (1985)
    • (1985) J. Math. Soc. Japan , vol.37 , pp. 391-413
    • Kanai, M.1
  • 15
    • 84972571397 scopus 로고
    • Rough isometries and the parabolicity of Riemannian manifolds
    • Kanai, M.: Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Japan, 38, 227-238 (1986)
    • (1986) J. Math. Soc. Japan , vol.38 , pp. 227-238
    • Kanai, M.1
  • 16
    • 0001437623 scopus 로고
    • Kontaktprobleme der konformen abbildung. Berichte verhande
    • Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte Verhande. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Klasse 88, 141-164 (1936)
    • (1936) Sächs. Akad. Wiss. Leipzig, Math.-Phys. Klasse , vol.88 , pp. 141-164
    • Koebe, P.1
  • 18
    • 0018457301 scopus 로고
    • A separator theorem for planar graphs
    • Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. of Appl. Math. 36, 177-189 (1979)
    • (1979) SIAM J. of Appl. Math. , vol.36 , pp. 177-189
    • Lipton, R.J.1    Tarjan, R.E.2
  • 19
    • 0002442697 scopus 로고
    • Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains
    • Lyons, T.: Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differential Geom. 26, 33-66 (1987)
    • (1987) J. Differential Geom. , vol.26 , pp. 33-66
    • Lyons, T.1
  • 20
    • 0010283607 scopus 로고
    • On Thurston's formulation and proof of Andreev's theorem
    • Computational Methods and Function Theory, (Ruscheweyh, Saff, Salinas, Varga eds.), Springer
    • Marden, A., Rodin, B.: On Thurston's formulation and proof of Andreev's theorem. In: Computational Methods and Function Theory, (Ruscheweyh, Saff, Salinas, Varga eds.), Lecture Notes in Mathematics, vol. 1435, Springer, 1989, pp. 103-115
    • (1989) Lecture Notes in Mathematics , vol.1435 , pp. 103-115
    • Marden, A.1    Rodin, B.2
  • 21
    • 84963048955 scopus 로고
    • Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces
    • Markvorsen, S., McGuinness, S., Thomassen, C.: Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces. Proc. London Math. Soc. 64, 1-20 (1992)
    • (1992) Proc. London Math. Soc. , vol.64 , pp. 1-20
    • Markvorsen, S.1    McGuinness, S.2    Thomassen, C.3
  • 22
    • 84968466613 scopus 로고
    • Geodesics and bounded harmonic functions on infinite planar graphs
    • Northshield, S.: Geodesics and bounded harmonic functions on infinite planar graphs. Proc. Amer. Math. Soc. 113, 229-233 (1991)
    • (1991) Proc. Amer. Math. Soc. , vol.113 , pp. 229-233
    • Northshield, S.1
  • 23
    • 84972500328 scopus 로고
    • The convergence of circle packings to the Riemann mapping
    • Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping, J. of Differential Geometry 26, 349-360 (1987)
    • (1987) J. of Differential Geometry , vol.26 , pp. 349-360
    • Rodin, B.1    Sullivan, D.2
  • 24
    • 0005828090 scopus 로고
    • Classification theory of Riemannian manifolds
    • Springer-Verlag
    • Sario, L., Nakai, M., Wang, C., Chung, L.O.: Classification Theory of Riemannian Manifolds, Lecture notes in mathematics vol. 605, Springer-Verlag, 1977, pp. 498
    • (1977) Lecture Notes in Mathematics , vol.605 , pp. 498
    • Sario, L.1    Nakai, M.2    Wang, C.3    Chung, L.O.4
  • 25
    • 51249167193 scopus 로고
    • Transboundary extremal length
    • Schramm, O.: Transboundary extremal length. J. d'Analyse Math., 66, 307-329 (1995)
    • (1995) J. d'Analyse Math. , vol.66 , pp. 307-329
    • Schramm, O.1
  • 26
    • 0030445434 scopus 로고    scopus 로고
    • Conformal uniformization and packings
    • Schramm, O.: Conformal uniformization and packings. Israel J. Math. 93, 399-428 (1996)
    • (1996) Israel J. Math. , vol.93 , pp. 399-428
    • Schramm, O.1
  • 27
    • 84968483712 scopus 로고
    • Rough isometries and Dirichlet finite harmonic functions on graphs
    • Soardi P.M.: Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. AMS 119, 1239-1248 (1993)
    • (1993) Proc. AMS , vol.119 , pp. 1239-1248
    • Soardi, P.M.1
  • 28
    • 0003195845 scopus 로고    scopus 로고
    • Potential theory on infinite networks
    • Springer, Berlin
    • Soardi, P.M.: Potential Theory on Infinite Networks, Lecture notes in mathematics, vol. 1590, Springer, Berlin, pp. 187
    • Lecture Notes in Mathematics , vol.1590 , pp. 187
    • Soardi, P.M.1
  • 29
    • 0002768694 scopus 로고
    • Random walks on infinite graphs and groups - A survey on selected topics
    • Woess, W.: Random walks on infinite graphs and groups - a survey on selected topics Bull. London Math. Soc. 26, 1-60 (1994)
    • (1994) Bull. London Math. Soc. , vol.26 , pp. 1-60
    • Woess, W.1
  • 30
    • 84972534160 scopus 로고
    • Parabolic and hyperbolic infinite networks
    • Yamasaki M.: Parabolic and hyperbolic infinite networks. Hiroshima Math. J. 7, 135-146 (1977)
    • (1977) Hiroshima Math. J. , vol.7 , pp. 135-146
    • Yamasaki, M.1
  • 31
    • 0010722686 scopus 로고
    • Discrete potentials on an infinite network
    • Yamasaki M.: Discrete potentials on an infinite network. Mem. Fac. Sci. Shimane Unif. 13, 31-44 (1979.)
    • (1979) Mem. Fac. Sci. Shimane Unif. , vol.13 , pp. 31-44
    • Yamasaki, M.1
  • 32
    • 84972565710 scopus 로고
    • Ideal boundary limit of discrete Dirichlet functions
    • Yamasaki, M.: Ideal boundary limit of discrete Dirichlet functions. Hiroshima Math. J. 16, 353-360 (1986)
    • (1986) Hiroshima Math. J. , vol.16 , pp. 353-360
    • Yamasaki, M.1


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