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LINEAR ISOPERIMETRIC INEQUALITY AND GROMOV HYPERBOLICITY ON ALEKSANDROV SURFACES

  • Oh, Byung-Geun (Department of Mathematics Education Hanyang University)
  • Received : 2010.04.21
  • Accepted : 2010.06.01
  • Published : 2010.06.30

Abstract

We prove that a simply-connected open Aleksandrov surface that satisfies a linear isoperimetric inequality is hyperbolic in the sense of Gromov.

Keywords

Acknowledgement

Supported by : Hanyang University

References

  1. A. D. Aleksandrov and V. A. Zalgaller, Intrinsic Geometry of Surfaces, AMS Transl. Math. Monographs, v. 15, Providence, RI, 1967.
  2. M. Bonk and A. Eremenko, Uniformly hyperbolic surfaces, Indiana Univ. Math. J. 49 (2000), no. 1, 61-80.
  3. C. Caratheodory, Uber die Begrenzung einfach zusammenhangender Gebiete, Math. Ann. 73, (1913), 323-370. https://doi.org/10.1007/BF01456699
  4. M. Coornaert, T. Delzant and A. Papadopoulos, Geometrie et theorie des groupes, LNM, Vol. 1441, Springer, Berlin, 1990.
  5. G. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monographs, Vol. 26, AMS, Providence, RI, 1969.
  6. M. Gromov, Hyperbolic Groups, In: Essays in Group theory (S. Gersten eds.), MSRI Publication 8, Springer, 1987, 75-263.
  7. E. Ghys and P. de la Harpe (eds.), Sur les Groupes Hyperbolique d'apres Mikhael Gromov, Birkhauser, Boston, MA, 1990.
  8. A. Huber, Zum potentialtheoretischen Aspekt der Alexandrowshen Flachentheorie, Comment. Math. Helv. 34 (1960), 99-126. https://doi.org/10.1007/BF02565931
  9. B.-G. Oh, Curvature and hyperbolicity on surfaces, Ph.D. thesis, Purdue University, 2004.
  10. Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, In: Geometry IV. Encyclopaedia of Mathematical Sciences (Yu. G. Reshetnyak eds.) Vol. 70, Springer, Berlin, 1993, 3-163.