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Classification of Riemannian manifolds in nonlinear potential theory. (English) Zbl 0771.53019

Authors’ summary: “The classification theory of Riemann surfaces is generalized to Riemannian \(n\)-manifolds in the conformally invariant case. This leads to the study of the existence of \(\mathcal A\)-harmonic functions of type \(n\) with various properties and to an extension of the definition of the classical notions with inclusions \(O_ G \subset O_{HP} \subset O_{HB} \subset O_{HD}\). In the classical case the properness of the inclusions were proved rather late, in the 50’s by Ahlfors and Tôki. Our main objective is to show that inclusions are proper also in the generalized case”.

MSC:

53C20 Global Riemannian geometry, including pinching
31C12 Potential theory on Riemannian manifolds and other spaces
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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[1] Ahlfors, L. V. and Sario, L.:Riemann Surfaces, Princeton University Press, Princeton, New Jersey (1960). · Zbl 0196.33801
[2] Aubin, T.:Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, Berlin-Heidelberg-New York (1982). · Zbl 0512.53044
[3] Donaldson, S. K. and Sullivan, D. P.: Quasiconformal 4-manifolds,Acta Math. 163 (1989), 181-252. · Zbl 0704.57008
[4] Epstein, C. L.: Positive harmonic functions on Abelian covers,Journal of Functional Analysis 82 (1989), 303-315. · Zbl 0685.31003
[5] Granlund, S., Lindqvist, P. and Martio, O.: Conformally invariant variational integrals,Trans. Amer. Math. Soc. 277 (1983), 43-73. · Zbl 0518.30024
[6] Heinonen, J. and Kilpeläinen, T.: 65-1-superharmonic functions and supersolutions of degenerate elliptic equations,Ark. Mat. 26 (1988), 87-105. · Zbl 0652.31006
[7] Heinonen, J., Kilpeläinen, T. and Martio, O.:Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford (1993). · Zbl 0780.31001
[8] Herron, D. A. and Koskela, P.: Uniform, Sobolev extension and quasiconformal circle domains,J. Analyse Math. 57 (1991), 172-202. · Zbl 0776.30014
[9] Hesse, J.: Ap-extremal length andp-capacity equality,Ark. Mat. 13 (1975), 131-144. · Zbl 0302.31009
[10] Holopainen, I.: Nonlinear potential theory and quasiregular mappings on Riemannian manifolds,Ann. Acad. Sci. Fenn. Ser. A I Math. Diss. 74 (1990). · Zbl 0698.31010
[11] Holopainen, I.: Positive solutions of quasilinear elliptic equations on Riemannian manifolds,Proc. London Math. Soc. (3)65 (1992), 651-672. · Zbl 0782.53030
[12] Holopainen, I. and Rickman, S.: A Picard type theorem for quasiregular mappings ofR n inton-manifolds with many ends,Rev. Mat. Iberoamericana 8 (1992), 131-148. · Zbl 0763.30006
[13] Holopainen, I. and Rickman, S.: Quasiregular mappings of the Heisenberg group,Math. Ann. 294 (1992), 625-643. · Zbl 0754.30018
[14] Iwaniec, T.:p-harmonic tensors and quasiregular mappings (to appear inAnn. of Math.). · Zbl 0785.30009
[15] Iwaniec, T. and Martin, G.: Quasiregular mappings in even dimensions (to appear inActa Math.). · Zbl 0785.30008
[16] Kanai, M.: Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds,J. Math. Soc. Japan 37 (1985), 391-413. · Zbl 0554.53030
[17] Lindqvist, P. and Martio, O.: Two theorems of N. Wiener for solutions of quasilinear elliptic equations,Acta Math. 155 (1985), 153-171. · Zbl 0607.35042
[18] Lyons, T.: Instability of Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains,J. Differential Geometry 26 (1987), 33-66. · Zbl 0599.60011
[19] Lyons, T. and McKean, H. P.: Winding of the plane Brownian motion,Adv. in Math. 51 (1984), 212-225. · Zbl 0541.60075
[20] Lyons, T. and Sullivan, D.: Function theory, random paths and covering spaces,J. Differential Geometry 19 (1984), 299-323. · Zbl 0554.58022
[21] Maz’ya, V. G.: On the continuity at a boundary point of solutions of quasi-linear elliptic equations,Vestnik Leningrad Univ. 3 (1976), 225-242 (English translation).
[22] McKean, H. P. and Sullivan, D.: Brownian motion and harmonic functions on the class surface of the thrice punctured sphere,Adv. in Math. 51 (1984), 203-211. · Zbl 0541.60076
[23] Phillips, A. and Sullivan, D.: Geometry of leaves,Topology 20 (1981), 209-218. · Zbl 0454.57016
[24] Reshetnyak, Yu. G.:Space Mappings with Bounded Distortion, Translations of Mathematical Monographs, Amer. Math. Soc., Vol. 73 (1989).
[25] Rickman, S.: On the number of omitted values of entire quasiregular mappings,J. Analyse Math. 37 (1980), 100-117. · Zbl 0451.30012
[26] Rickman, S.: A defect relation for quasimeromorphic mappings,Ann. of Math. 114 (1981), 165-191. · Zbl 0456.30017
[27] Rickman, S.: Quasiregular mappings and metrics on then-sphere with punctures,Comment. Math. Helv. 59 (1984), 134-148. · Zbl 0556.30016
[28] Rickman, S.: The analogue of Picard’s theorem for quasiregular mappings in dimension three,Acta Math. 154 (1985), 195-242. · Zbl 0617.30024
[29] Rickman, S.: Defect relation and its realization for quasiregular mappings, preprint, University of Helsinki (1992). · Zbl 0764.30017
[30] Rickman, S.:Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag (to appear). · Zbl 0816.30017
[31] Sario, L.et al.:Classification Theory of Riemannian Manifolds, Lecture Notes in Math., vol. 605, Springer-Verlag, Berlin-Heidelberg-New York (1977). · Zbl 0355.31001
[32] Sario, L. and Nakai, M.:Classification Theory of Riemann Surfaces, Springer-Verlag, Berlin-Heidelberg-New York (1970). · Zbl 0199.40603
[33] Serrin, J.: Local behavior of solutions of quasilinear equations,Acta Math. 111 (1964), 247-302. · Zbl 0128.09101
[34] Tanaka, H.: Harmonic boundaries of Riemannian manifolds,Nonlinear Analysis, Theory, Methods & Applications 14 (1990), 55-67. · Zbl 0712.31004
[35] Trudinger, N. S.: On Harnack type inequalities and their applications to quasilinear elliptic equations,Comm. Pure Appl. Math. 20 (1967), 721-747. · Zbl 0153.42703
[36] Vuorinen, M.:Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math., vol. 1319, Springer-Verlag, Berlin-Heidelberg-New York (1988). · Zbl 0646.30025
[37] Väisälä, J.:Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin-Heidelberg-New York (1971). · Zbl 0221.30031
[38] Zeidler, E.:Nonlinear Functional Analysis and Its Applications II/B, Springer-Verlag, Berlin-Heidelberg-New York (1990). · Zbl 0684.47029
[39] Ziemer, W. P.: Extremal length andp-capacity,Michigan Math. J. 16 (1969), 43-51. · Zbl 0172.38701
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