On the conformal type of a Riemannian manifold

Functional Analysis and its Applications

Volumn 30, Issue 2, 1996, Pages 106-117

  Zorich, V A ^a   Kesel'man, V M ^a  

^a NONE

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[No Author keywords available]

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EID: 0039786420     PISSN: 00162663     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02509450     Document Type: Article

References (50)

1. L. Ahlfors, "Sur le type d'une surface de Riemann," C. R. Acad. Sci. Paris Ser. A, 201, 30-32 (1935).
2. L. Ahlfors, "Zur Theorie des Überlagerungsflächen," Acta Math., 65, 157-194 (1935).
3. L. Ahlfors, Conformal Invariants (Topics in Geometric Function Theory), McGraw-Hill, New York (1973).
4. L. Ahlfors and A. Beurling, "Conformal invariants and function-theoretic null-sets," Acta Math., 83, 101-129 (1950).
5. L. Ahlfors and L. Sario, Riemann Surfaces, Princeton Univ. Press, Princeton, N.J. (1960).
6. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, "Conformal invariants, quasiconformal maps and special functions," in: Lect. Notes Math., Vol. 1508 (1992), pp. 1-19.
7. P. Caraman, n-Dimensional Quasiconformal Mappings, Editura Academiei Romane, Bucharest, Abacus Press, Tunbridge Wells Haessner Publishing, Inc., Newfoundland, N.J. (1974).
8. S. V. Cheng and S. T. Yau, "Differential equations on Riemannian manifolds and their geometric applications," Commun. Pure Appl. Math., 28, 333-354 (1975).
9. M. H. Freedman and Z. H. He, "Divergence free fields: Energy and asymptotic crossing number," Ann. Math., 134, 189-229 (1991).
10. M. H. Freedman and Z. H. He, "Links of tori and the energy of incompressible flows," Topology, 30, No. 2, 283-287 (1991).
11. J. Lelong-Ferrand, "Invariant conform globaux sur les variétés Riemanniennes," J. Diff. Geom., 8, 487-510 (1973).
12. J. Ferrand, "Conformal capacity and conformally invariant metrics," to appear.
13. J. Ferrand, "Conformal capacity and conformally invariant functions on manifolds," C. R. Acad. Sci. Paris Sér. I, 218, 213-216 (1994).
14. B. Fuglede, "Extremal length and functional completion," Acta Math., 83, 101-129 (1957).
15. F. V. Gehring, "Symmetrization of rings in space," Trans. Am. Math. Soc., 101, 499-519 (1961).
16. F. V. Gehring, "Extremal length definitions for the conformal capacity of rings in space," Mich. Math. J., 9, 137-150 (1962).
17. F. V. Gehring, "Topics in Quasiconformal Mappings," in: Lect. Notes Math., Vol. 1508 (1992), pp. 20-38;
18. Reprinted from Proc. Internat. Congr. Math. (Berkeley, California 1986), Vol. 1,
19. Am. Math. Soc., 1987, pp. 62-80.
20. A. A. Grigoryan, "The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds," Mat. Sb., 128, No. 3, 354-363 (1985).
21. M. Gromov, "Hyperbolic manifolds, groups and actions," in: Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Studies, Vol. 97, Princeton Univ. Press, Princeton, N.J. (1981), pp. 183-213.
22. M. Gromov, Structures Métriques pour les Variétés Riemanniennes, Notes du cours rédigé par J. Lafontaine et P. Pansu, CEDIC/Fernand-Nathan et Soc. Math, de France, Paris (1981).
23. R. Grimaldi and P. Pansu, "Sur la croissance du volume dans une classe conforme," J. Math. Pures Appl., (9) 71, No. 1, 1-19 (1992).
24. V. M. Gol'dstein and S. K. Vodop'yanov, "Metric completion of a domain by means of a conformal capacity that is invariant under quasiconformal mappings," Dokl. Acad. Nauk SSSR, 238, No, 5, 1040-1042 (1978);
25. English transl. in Soviet Math. Dokl., 19, 158-161 (1978).
26. J. Hesse, "A p-extremal length and p-capacity equality," Ark. Mat., 13, 131-144 (1975).
27. S. Hildebrandt, "Liouville theorems for harmonic mappings and approach to Bernstein theorems," Ann. Math. Stud:, 102, 107-131 (1982).
28. I. Holopainen, Nonlinear Potential Theory and Quasiregular Mappings on Riemannian Manifolds, Ann. Acad. Sci. Fenn. Ser. AI, Math. Dissert., Vol. 74, Helsinki (1990).
29. L. Karp, "Subharmonic functions on real and complex manifolds," Math. Z., 178, 535-554 (1982).
30. V. M. Keselman, "Riemannian manifolds of α-parabolic type," Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 4, 81-83 (1985).
31. T. Kilpelainen, "Potential theory for supersolutions of degenerate elliptic equations," Indiana Univ. Math. J., 38, No. 2, 253-275 (1989).
32. C. Loewner, "On the conformal capacity in space," J. Math. Mech., 8, 411-414 (1959).
33. J. Milnor, "A note on curvature and fundamental group," J. Diff. Geom., 2, 1-7 (1968).
34. J. Milnor, "On deciding whether a surface is parabolic or hyperbolic," Am. Math. Monthly, 84, No. 1, 43-46 (1977).
35. V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-Tokyo (1985).
36. V. M. Miklyukov, "Conformal type of surfaces, Liouville's theorem, and Bernstein's theorem," Dokl. Akad. Nauk SSSR, 242, No. 3, 537-540 (1978).
37. G. D. Mostow, "Quasiconformal mappings in n-space and the rigidity of hyperbolic space forms," IHES Publ. Math., 34, 53-104 (1968).
38. R. Osserman, "Isoperimetric inequalities," Bull. Am. Math. Soc., 84, 1182-1238 (1978).
39. P. Pansu, "An isoperimetric inequality on the Heisenberg group," Proceedings of "Differential Geometry and Homogeneous Spaces," Torino (1983), pp. 159-174.
40. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Monographs, 73, Am. Math. Soc., Providence, R.I. (1989).
41. S. Rickman, Quasiregular Mappings, Springer-Verlag (1993).
42. E. Schlesinger, "Conformal invariants and prime ends," Am. J. Math., 80, 83-102 (1958).
43. L. Sario, M. Nakai, C. Wang, and L. O. Chung, Classification Theory of Riemannian Manifolds, Lect. Notes Math., Vol. 605, Springer-Verlag (1977).
44. A. S. Shvarts (Schwarz), "A volume invariant of coverings," Dokl. Akad. Nauk SSSR, 105, 32-34 (1955).
45. H. Tanaka, "Harmonic boundaries of Riemannian manifolds," Nonlinear Analysis, Methods and Applications, 14, No. 1, 55-67 (1990).
46. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lect. Notes Math., Vol. 229, Springer-Verlag (1971).
47. M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lect. Notes Math., Vol. 1319, Springer-Verlag (1988).
48. W. P. Ziemer, "Extremal length as a capacity," Mich. Math. J., 17, 117-128 (1970).
49. V. A. Zorich, "On Gromov's geometric version of the global homeomorphism theorem for quasiconformal mappings," in: XIV Rolf Nevanlinna Colloquium, Helsinki, June 10-14, 1990, Abstracts, p. 36.
50. V. A. Zorich, "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems," in: Lect. Notes Math., Vol. 1508 (1992), pp. 131-148.