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Aleksandrov surfaces and hyperbolicity

Author: Byung-Geun Oh
Journal: Trans. Amer. Math. Soc. 357 (2005), 4555-4577
MSC (2000): Primary 30F20, 30D30; Secondary 28A75, 30D35
Published electronically: June 10, 2005
MathSciNet review: 2156721
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Abstract: Aleksandrov surfaces are a generalization of two-dimensional Riemannian manifolds, and it is known that every open simply-connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). We prove a criterion for hyperbolicity of Aleksandrov surfaces which have nice tilings and where negative curvature dominates. We then apply this to generalize a result of Nevanlinna and give a partial answer for his conjecture about line complexes.

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  • 1. L. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973. MR 0357743 (50:10211)
  • 2. A. D. Aleksandrov, Isoperimetric inequalities for curved surfaces, C. R. (Doklady) Acad. Sci. USSR (N.S.) 47 (1945), pp. 235-238. MR 0013543 (7:167f)
  • 3. A. D. Aleksandrov, Über eine Verallgemeinerung der Riemannschen Geometrie, Schr. Forschungsinst. Math. 1 (1957), pp. 33-84. MR 0087119 (19:304h)
  • 4. A. D. Aleksandrov and V. A. Zalgaller, Intrinsic Geometry of Surfaces, AMS Transl. Math. Monographs, Vol. 15, Providence, RI, 1967. MR 0216434 (35:7267)
  • 5. I. Benjamini, S. Merenkov and O. Schramm, A negative answer to Nevanlinna's type question and a parabolic surface with a lot of negative curvature, Proc. Amer. Math. Soc. 132 (2004), no. 3, pp. 641-647 MR 2019938 (2004k:30096)
  • 6. V. N. Berestovskij and I. G. Nikolaev, Multidimensional Generalized Riemannian Spaces, In: Geometry IV. Encyclopaedia of Mathematical Sciences (Yu. G. Reshetnyak ed.) Vol. 70, Springer, Berlin, 1993, pp. 165-243. MR 1263965
  • 7. M. Bonk, Singular Surfaces and Meromorphic Functions, Notices of AMS 49 (2002), no. 6, pp. 647-657. MR 1908328
  • 8. M. Bonk and A. Eremenko, Schlicht regions for entire and meromorphic functions, J. Analyse Math. 77 (1999), pp. 69-104. MR 1753483 (2002c:30037)
  • 9. M. Bonk and A. Eremenko, Uniformly hyperbolic surfaces, Indiana Univ. Math. J. 49 (2000) no. 1, pp. 61-80. MR 1777037 (2001g:53120)
  • 10. M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann. of Math. (2) 152 (2000) no. 2, pp. 551-592. MR 1804531 (2002a:30050)
  • 11. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, Berlin, 1988. MR 0936419 (89b:52020)
  • 12. W. Chauvenet, A Treatise on Plane and Spherical Trigonometry, J. B. Lippincott Co., Philadelphia, 1850.
  • 13. M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, LNM, Vol. 1441, Springer, Berlin, 1990. MR 1075994 (92f:57003)
  • 14. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. MR 0164038 (29:1337)
  • 15. A. Huber, Zum potentialtheoretischen Aspekt der Alexandrowshen Flächentheorie, Comment. Math. Helv. 34 (1960), pp. 99-126. MR 0115147 (22:5949)
  • 16. R. Nevanlinna, Analytic Functions, Springer-Verlag, New York-Berlin, 1970. MR 0279280 (43:5003)
  • 17. B. Oh, Linear Isoperimetric Inequality, Gromov Hyperbolicity and Riemann Surfaces of class $F_q$, preprint.
  • 18. Yu. G. Reshetnyak, Two-Dimensional Manifolds of Bounded Curvature, In: Geometry IV. Encyclopaedia of Mathematical Sciences (Yu. G. Reshetnyak ed.) Vol. 70, Springer, Berlin, 1993, pp. 3-163. MR 1263964
  • 19. S. Stoilov, Leçons sur les principes topologiques de la théorie des fonctions analytiques, Gauthier-Villars, Paris, 1956.
  • 20. O. Teichmüller, Untersuchungen über konforme und quasikonforme Abbildung, Dtsch. Math. 3, 1938, no. 6, pp. 621-678.
  • 21. J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, LNM, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009 (56:12260)

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Additional Information

Byung-Geun Oh
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350

Received by editor(s): December 17, 2003
Published electronically: June 10, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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