Volume growth, Green’s functions, and parabolicity of ends
Ilkka Holopainen
Source: Duke Math. J. Volume 97, Number 2
(1999), 319-346.
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References
[Ab] Uwe Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 651–670.
Mathematical Reviews (MathSciNet): MR87j:53058
Zentralblatt MATH: 0595.53043
[AG] Uwe Abresch and Detlef Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), no. 2, 355–374.
Mathematical Reviews (MathSciNet): MR91a:53071
Zentralblatt MATH: 0704.53032
Digital Object Identifier: doi:10.2307/1990957
JSTOR: links.jstor.org
[Ah1] L. V. Ahlfors, Sur le type d'une surface de Riemann, C. R. Acad. Sci. Paris Sér. A 201 (1935), 30–32.
Zentralblatt MATH: 0011.40701
[Ah2] L. V. Ahlfors, Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157–194.
Zentralblatt MATH: 0012.17204
[BC] Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York, 1964.
Mathematical Reviews (MathSciNet): MR29:6401
Zentralblatt MATH: 0132.16003
[B] Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230.
Mathematical Reviews (MathSciNet): MR84e:58076
Zentralblatt MATH: 0501.53030
[C] Mingliang Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 371–377.
Mathematical Reviews (MathSciNet): MR92f:53045
Zentralblatt MATH: 0728.53026
Digital Object Identifier: doi:10.1090/S0273-0979-1991-16038-6
Project Euclid: euclid.bams/1183656877
[CG] Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128.
Mathematical Reviews (MathSciNet): MR46:2597
Zentralblatt MATH: 0223.53033
Project Euclid: euclid.jdg/1214430220
[CGT] Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), no. 1, 15–53.
Mathematical Reviews (MathSciNet): MR84b:58109
Zentralblatt MATH: 0493.53035
Project Euclid: euclid.jdg/1214436699
[CY] S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354.
Mathematical Reviews (MathSciNet): MR52:6608
Zentralblatt MATH: 0312.53031
Digital Object Identifier: doi:10.1002/cpa.3160280303
[CSC1] Thierry Coulhon and Laurent Saloff-Coste, Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamericana 11 (1995), no. 3, 687–726.
Mathematical Reviews (MathSciNet): MR96m:53035
Zentralblatt MATH: 0845.58054
[CSC2] T. Coulhon and L. Saloff-Coste, Harnack inequality and hyperbolicity for the $p$-Laplacian with applications to quasiregular mappings, preprint.
[G1] A. Grigor'yan, On the existence of a Green function on a manifold, Uspekhi Mat. Nauk 38 (1983), 161–162, (in Russian); English transl. in Russian Math. Surveys 38 (1983), 190–191.
Zentralblatt MATH: 0542.35025
Mathematical Reviews (MathSciNet): MR693728
[G2] A. Grigor'yan, On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds, Mat. Sb. (N.S.) 128 (1985), 354–363, (in Russian); English transl. in Math. USSR-Sb. 56 (1987), 349–358.
Zentralblatt MATH: 0596.31004
Mathematical Reviews (MathSciNet): MR815269
[G3] A. Grigor'yan, The heat equation on noncompact Riemannian manifolds, Math. USSR-Sb. 72 (1992), 47–77.
Zentralblatt MATH: 0776.58035
Mathematical Reviews (MathSciNet): MR1098839
[HK] Piotr Hajłasz and Pekka Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 10, 1211–1215.
Mathematical Reviews (MathSciNet): MR96f:46062
Zentralblatt MATH: 0837.46024
[HKM] Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1993.
Mathematical Reviews (MathSciNet): MR94e:31003
Zentralblatt MATH: 0780.31001
[H1] Ilkka Holopainen, Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes (1990), no. 74, 45.
Mathematical Reviews (MathSciNet): MR91e:31029
Zentralblatt MATH: 0698.31010
[H2] Ilkka Holopainen, Positive solutions of quasilinear elliptic equations on Riemannian manifolds, Proc. London Math. Soc. (3) 65 (1992), no. 3, 651–672.
Mathematical Reviews (MathSciNet): MR94d:58161
Zentralblatt MATH: 0739.53030
Digital Object Identifier: doi:10.1112/plms/s3-65.3.651
[H3] Ilkka Holopainen, Solutions of elliptic equations on manifolds with roughly Euclidean ends, Math. Z. 217 (1994), no. 3, 459–477.
Mathematical Reviews (MathSciNet): MR95j:58179
Zentralblatt MATH: 0833.58036
Digital Object Identifier: doi:10.1007/BF02571955
[HR] Ilkka Holopainen and Seppo Rickman, Ricci curvature, Harnack functions, and Picard type theorems for quasiregular mappings, Analysis and topology eds. C. Cazacu, O. Lehto, and T. Rassias, World Sci. Publishing, River Edge, NJ, 1998, pp. 315–326.
Mathematical Reviews (MathSciNet): MR99j:30026
Zentralblatt MATH: 0945.30022
[J] David Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), no. 2, 503–523.
Mathematical Reviews (MathSciNet): MR87i:35027
Zentralblatt MATH: 0614.35066
Digital Object Identifier: doi:10.1215/S0012-7094-86-05329-9
Project Euclid: euclid.dmj/1077305054
[K] Masahiko Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391–413.
Mathematical Reviews (MathSciNet): MR87d:53082
Zentralblatt MATH: 0554.53030
Digital Object Identifier: doi:10.2969/jmsj/03730391
Project Euclid: euclid.jmsj/1230395497
[K1] Atsushi Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 593–622.
Mathematical Reviews (MathSciNet): MR90d:53049
Zentralblatt MATH: 0662.53032
[K2] Atsushi Kasue, Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature. I, Geometry and analysis on manifolds (Katata/Kyoto, 1987), Lecture Notes in Math., vol. 1339, Springer, Berlin, 1988, pp. 158–181.
Mathematical Reviews (MathSciNet): MR89i:53030
Zentralblatt MATH: 0685.31004
Digital Object Identifier: doi:10.1007/BFb0083054
[Ke] V. M. Kesel'man, Riemannian manifolds of $\alpha$-parabolic type, Izv. Vyssh. Uchebn. Zaved. Mat. (1985), no. 4, 81–83, 88, (in Russian).
Mathematical Reviews (MathSciNet): MR86m:31009
Zentralblatt MATH: 0583.53037
[KZ] V. A. Zorich and V. M. Kesel'man, On the conformal type of a Riemannian manifold, Funktsional. Anal. i Prilozhen. 30 (1996), no. 2, 40–55, 96, (in Russian), English transl. in Functional Anal. Appl. 30 (1996), 106–117.
Mathematical Reviews (MathSciNet): MR97f:31022
Zentralblatt MATH: 0873.53025
[KM] Tero Kilpeläinen and Jan Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), no. 1, 137–161.
Mathematical Reviews (MathSciNet): MR95a:35050
Zentralblatt MATH: 0820.35063
Digital Object Identifier: doi:10.1007/BF02392793
[Li] P. Li, Curvature and function theory on Riemannian manifolds, preprint.
Mathematical Reviews (MathSciNet): MR1919432
Zentralblatt MATH: 1066.53084
[LT1] Peter Li and Luen-Fai Tam, Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set, Ann. of Math. (2) 125 (1987), no. 1, 171–207.
Mathematical Reviews (MathSciNet): MR88m:58039
Zentralblatt MATH: 0622.58001
Digital Object Identifier: doi:10.2307/1971292
JSTOR: links.jstor.org
[LT2] Peter Li and Luen-Fai Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), no. 2, 359–383.
Mathematical Reviews (MathSciNet): MR93b:53033
Zentralblatt MATH: 0768.53018
Project Euclid: euclid.jdg/1214448079
[LT3] Peter Li and Luen-Fai Tam, Green's functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), no. 2, 277–318.
Mathematical Reviews (MathSciNet): MR96f:53054
Zentralblatt MATH: 0827.53033
Project Euclid: euclid.jdg/1214456219
[LY] Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201.
Mathematical Reviews (MathSciNet): MR87f:58156
Zentralblatt MATH: 0611.58045
Digital Object Identifier: doi:10.1007/BF02399203
[LM] P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), no. 3-4, 153–171.
Mathematical Reviews (MathSciNet): MR87g:35074
Zentralblatt MATH: 0607.35042
Digital Object Identifier: doi:10.1007/BF02392541
[L] Zhong-dong Liu, Ball covering property and nonnegative Ricci curvature outside a compact set, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 459–464.
Mathematical Reviews (MathSciNet): MR94b:53075
Zentralblatt MATH: 0788.53028
[M1] V. G. Maz'ya, On the continuity at a boundary point of solutions of quasilinear elliptic equations, Vestnik Leningrad. Univ. Mat. 3 (1976), 225–242.
[M2] Vladimir G. Maz'ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
Mathematical Reviews (MathSciNet): MR87g:46056
Zentralblatt MATH: 0692.46023
[RSV] Marco Rigoli, Maura Salvatori, and Marco Vignati, A note on $p$-subharmonic functions on complete manifolds, Manuscripta Math. 92 (1997), no. 3, 339–359.
Mathematical Reviews (MathSciNet): MR98e:53068
Zentralblatt MATH: 0873.31012
Digital Object Identifier: doi:10.1007/BF02678198
[SC1] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices (1992), no. 2, 27–38.
Mathematical Reviews (MathSciNet): MR93d:58158
Zentralblatt MATH: 0769.58054
Digital Object Identifier: doi:10.1155/S1073792892000047
[SC2] Laurent Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), no. 2, 417–450.
Mathematical Reviews (MathSciNet): MR93m:58122
Zentralblatt MATH: 0735.58032
Project Euclid: euclid.jdg/1214448748
[S] James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.
Mathematical Reviews (MathSciNet): MR30:337
Zentralblatt MATH: 0128.09101
Digital Object Identifier: doi:10.1007/BF02391014
[V1] Nicholas Th. Varopoulos, The Poisson kernel on positively curved manifolds, J. Funct. Anal. 44 (1981), no. 3, 359–380.
Mathematical Reviews (MathSciNet): MR84h:58142a
Zentralblatt MATH: 0507.58046
Digital Object Identifier: doi:10.1016/0022-1236(81)90015-X
[V2] N. Th. Varopoulos, Green's functions on positively curved manifolds, J. Funct. Anal. 45 (1982), no. 1, 109–118.
Mathematical Reviews (MathSciNet): MR84h:58142b
Zentralblatt MATH: 0497.58020
Digital Object Identifier: doi:10.1016/0022-1236(82)90007-6
[V3] N. T. Varopoulos, Potential theory and diffusion on Riemannian manifolds, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 821–837.
Mathematical Reviews (MathSciNet): MR85a:58103
Zentralblatt MATH: 0558.31009
[Vu] Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988.
Mathematical Reviews (MathSciNet): MR89k:30021
Zentralblatt MATH: 0646.30025
[Y] Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228.
Mathematical Reviews (MathSciNet): MR55:4042
Zentralblatt MATH: 0291.31002
Digital Object Identifier: doi:10.1002/cpa.3160280203
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