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Bi-Lipschitz-equivalent Aleksandrov surfaces, I


Authors: A. Belen'kii and Yu. Burago
Translated by: Yu. D. Burago
Original publication: Algebra i Analiz, tom 16 (2004), nomer 4.
Journal: St. Petersburg Math. J. 16 (2005), 627-638
MSC (2000): Primary 53C45
DOI: https://doi.org/10.1090/S1061-0022-05-00869-1
Published electronically: June 21, 2005
MathSciNet review: 2090849
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Abstract | References | Similar Articles | Additional Information

Abstract: In this first paper of two, it is proved that two compact Aleksandrov surfaces with bounded integral curvature and without peak points are bi-Lipschitz-equivalent if they are homeomorphic. Also, conditions under which two tubes with finite negative part of integral curvature are bi-Lipschitz-equivalent are considered. In the second paper an estimate depending only on several geometric characteristics is found for a bi-Lipschitz constant.


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  • [AZ] A. D. Aleksandrov and V. A. Zalgaller, Two-dimensional manifolds of bounded curvature. (Foundations of the intrinsic geometry of surfaces), Trudy Mat. Inst. Steklov. 63 (1962), 162 pp.; English transl., Intrinsic geometry of surfaces, Transl. Math. Monogr., vol. 15, Amer. Math. Soc., Providence, RI, 1967. MR 0151930 (27:1911); MR 0216434 (35:7267)
  • [Bak] I. Ya. Bakel'man, Chebyshev networks in manifolds of bounded curvature, Trudy Mat. Inst. Steklov. 76 (1965), 124-129; English transl. in Proc. Steklov Inst. Math. No. 76 (1965) (1967). MR 0209988 (35:883)
  • [BB] Yu. D. Burago and S. V. Buyalo, Metrics of upper bounded curvature on 2-polyhedra. II, Algebra i Analiz 10 (1998), no. 4, 62-112; English transl., St. Petersburg Math. J. 10 (1999), no. 4, 619-650. MR 1654071 (99j:53086)
  • [BL] M. Bonk and U. Lang, Bi-Lipschitz parametrization of surfaces, Math. Ann. 327 (2003), 135-169. (Abstract DOI 10.1007/s00208-003-0443-8). MR 2006006 (2004i:53100)
  • [Gr] D. Grieser, Quasiisometry of singular metrics, Houston J. Math. 28 (2002), 741-752 (electronic). MR 1953683 (2004b:53043)
  • [Hub] A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13-72. MR 0094452 (20:970)
  • [Resh] Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, Geometry, 4, Itogi Nauki i Tekhniki Ser. Sovrem. Probl. Mat. Fund. Naprav., t. 70, VINITI, Moscow, 1989, pp. 7-189; English transl., Encyclopaedia Math. Sci., vol. 70, Springer-Verlag, Berlin, 1993, pp. 3-163. MR 1099202 (92b:53104); MR 1263964
  • [RS] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergeb. Math. Grenzgeb., vol. 69, Springer, Berlin etc., 1972. MR 0350744 (50:3236)
  • [Ver] A. Verner, A condition for the finite connectedness of complete nonclosed surfaces, Leningrad. Gos. Ped. Inst. Uchen. Zap. 395 (1970), 100-131. (Russian) MR 0287489 (44:4693)

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Additional Information

A. Belen'kii
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Yu. Burago
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: yuburago@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-05-00869-1
Keywords: Two-dimensional manifold of bounded integral curvature, Lipschitz metric, comparison triangle.
Received by editor(s): September 29, 2003
Published electronically: June 21, 2005
Additional Notes: The second author was partly supported by grants RFBR 02-01-00090, SS-1914.2003.1, and CRDF RM1-2381-ST-02
Article copyright: © Copyright 2005 American Mathematical Society

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