|
Aleksandrov surfaces and hyperbolicity
Author(s):
Byung-Geun
Oh
Journal:
Trans. Amer. Math. Soc.
357
(2005),
4555-4577.
MSC (2000):
Primary 30F20, 30D30;
Secondary 28A75, 30D35
Posted:
June 10, 2005
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 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
 , 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
 -dimensional quasiconformal mappings, LNM, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009 (56:12260)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
30F20, 30D30,
28A75, 30D35
Retrieve articles in all Journals with MSC
(2000):
30F20, 30D30,
28A75, 30D35
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
Email:
boh@math.purdue.edu, bgoh@math.washington.edu
DOI:
10.1090/S0002-9947-05-03977-2
PII:
S 0002-9947(05)03977-2
Received by editor(s):
December 17, 2003
Posted:
June 10, 2005
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org