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Dirichlet finite harmonic functions and points at infinity of graphs and manifolds. (English) Zbl 1145.53310

Summary: We consider the Royden compactifications relative to \(p\)-Dirichlet integrals of infinite graphs and noncompact Riemannian manifolds, and study the behavior of rough isometries in the compactifications, proving bijective correspondence of the spaces of \(p\)-harmonic functions with finite \(p\)-energy.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58D17 Manifolds of metrics (especially Riemannian)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

[1] T. Coulhon and L. Saloff-Coste, Variétés riemanniennes isométriques à l’infini, Rev. Mat. Iberoamericana 11 (1995), no. 3, 687-726. · Zbl 0845.58054
[2] T. Coulhon, I. Holopainen and L. Saloff-Coste, Harnack inequality and hyperbolicity for subelliptic \(p\)-Laplacians with applications to Picard type theorems, Geom. Funct. Anal. 11 (2001), no. 6, 1139-1191. · Zbl 1005.58013
[3] J. Eells and B. Fuglede, Harmonic maps between Riemannian polyhedra , Cambridge Univ. Press, Cambridge, 2001. · Zbl 0979.31001
[4] P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp. · Zbl 0954.46022
[5] I. Holopainen, Rough isometries and \(p\)-harmonic functions with finite Dirichlet integral, Rev. Mat. Iberoamericana 10 (1994), no. 1, 143-176. · Zbl 0797.31008
[6] I. Holopainen and P. M. Soardi, \(p\)-harmonic functions on graphs and manifolds, Manuscripta Math. 94 (1997), no. 1, 95-110. · Zbl 0898.31007
[7] M. Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391-413. · Zbl 0554.53030
[8] M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986), no. 2, 227-238. · Zbl 0577.53031
[9] M. Kanai, Analytic inequalities, and rough isometries between noncompact Riemannian manifolds, in Curvature and topology of Riemannian manifolds (Katata, 1985) , 122-137, Lecture Notes in Math., 1201, Springer, Berlin. · Zbl 0593.53026
[10] A. Kasue, Convergence of metric graphs and energy forms. (Preprint). · Zbl 1196.31004
[11] K. Kuwae, Y. Machigashira and T. Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269-316. · Zbl 1001.53017
[12] Y. H. Lee, Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds, Math. Ann. 318 (2000), no. 1, 181-204. · Zbl 0968.58018
[13] Y. H. Lee, Rough isometry and \(p\)-harmonic boundaries of complete Riemannian manifolds, Potential Anal. 23 (2005), no. 1, 83-97. · Zbl 1082.31005
[14] M. Nakai, Potential theory on Royden compactifications, Bull. Nagoya Inst. Tech. 47 (1995), 171-191. (in Japanese). · Zbl 0849.31016
[15] M. Nakai, Existence of quasi-isometric mappings and Royden compactifications, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 1, 239-260. · Zbl 0947.30010
[16] H. Tanaka, Harmonic boundaries of Riemannian manifolds, Nonlinear Anal. 14 (1990), no. 1, 55-67. · Zbl 0712.31004
[17] L. Sario and M. Nakai, Classification theory of Riemann surfaces , Springer, New York, 1970. · Zbl 0199.40603
[18] P. M. Soardi, Potential theory on infinite networks , Lecture Notes in Math., 1590, Springer, Berlin, 1994. · Zbl 0818.31001
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