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St. Petersburg Mathematical Journal

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Bi-Lipschitz-equivalent Aleksandrov surfaces, II

Author: Yu. Burago
Translated by: the author
Original publication: Algebra i Analiz, tom 16 (2004), nomer 6.
Journal: St. Petersburg Math. J. 16 (2005), 943-960
MSC (2000): Primary 53C45
Published electronically: November 17, 2005
MathSciNet review: 2117448
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Abstract: It is proved that any two homeomorphic closed Aleksandrov surfaces of bounded integral curvature are bi-Lipschitz-equivalent with constant depending only on their Euler number, upper bounds for their diameters and negative integral curvatures, and two positive numbers $ \varepsilon$ and $ l$ such that each loop of length at most $ l$ bounds a disk of positive curvature at most $ 2\pi-\varepsilon$.

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  • [AZ] A. D. Aleksandrov and V. A. Zalgaller, Two-dimensional manifolds of bounded curvature, Trudy Mat. Inst. Steklov. 63 (1962), 262 pp.; English transl., Intrinsic geometry of surfaces, Transl. Math. Monogr., vol. 15, Amer. Math. Soc., Providence, RI, 1967, 327 pp. 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. 1967. MR 0209988 (35:883)
  • [BeBu] A. Belen'kii and Yu. Burago, Bi-Lipschitz-equivalent surfaces. I, Algebra i Analiz 16 (2004), no. 4, 24-40; English transl. in St. Petersburg Math. J. 16 (2005), no. 4. MR 2090849
  • [BL] M. Bonk and U. Lang, Bi-Lipschitz parametrization of surfaces, Math. Ann. 327 (2003), 135-169; (DOI: 10.1007/s00208-003-0443-8). MR 2006006 (2004i:53100)
  • [B] Yu. Burago, Isometric imbedding of a manifold of bounded curvature into Euclidean space, Leningrad. Gos. Ped. Inst. Uchen. Zap. 395 (1970), 48-86. (Russian) MR 0303473 (46:2610)
  • [BZ] Yu. Burago and V. Zalgaller, Polyhedral embedding of a net, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1960, vyp. 2, 66-80. (Russian) MR 0116317 (22:7112)
  • [BZ1] -, Isometric piecewise linear immersions of two-dimensional manifolds with polyhedral metrics into $ \Bbb R^3$, Algebra i Analiz 7 (1995), no. 3, 76-95; English transl., St. Petersburg Math. J. 7 (1996), no. 3, 369-385. MR 1353490 (96g:53091)
  • [I] S. V. Ivanov, Gromov-Hausdorff convergence and volumes of manifolds, Algebra i Analiz 9 (1997), no. 5, 65-83; English transl., St. Petersburg Math. J. 9 (1998), no. 5, 945-959. MR 1604389 (98k:53052)
  • [P] P. Petersen, A finiteness theorem for metric spaces, J. Differential Geom. 31 (1990), 387-395. MR 1037407 (91d:53070)
  • [Resh] Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, Geometry, 4, Itogi Nauki i Tekhniki Ser. Sovrem. Probl. Mat. Fund. Naprav., vol. 70, VINITI, Moscow, 1989, pp. 7-189; English transl., Encyclopaedia Math. Sci., vol. 70, Springer, Berlin, 1993, pp. 3-163. MR 1099202 (92h:53104); MR 1263964
  • [Resh1] -, Investigation of manifolds of bounded curvature in terms of isothermic coordinates, Izv. Sibirsk. Otdel. Akad. Nauk SSSR 1959, no. 10, 15-28. (Russian)
  • [Sh] T. Shioya, The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature, Trans. Amer. Math. Soc. 351 (1999), 1765-1801. MR 1458311 (99h:53052)

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Additional Information

Yu. Burago
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Bi-Lipschitz map, two-dimensional manifold, bounded integral curvature, Aleksandrov surface
Received by editor(s): March 16, 2004
Published electronically: November 17, 2005
Additional Notes: This work was partially supported by grants RFBR 02-01-00090, SS-1914.2003.1, CRDF RM1-2381-ST-02, and by the Shapiro Foundation of Pennsylvania State University
Article copyright: © Copyright 2005 American Mathematical Society

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