Frontiers of Mathematics in China

, Volume 14, Issue 2, pp 349–380 | Cite as

Lipschitz Gromov-Hausdorff approximations to two-dimensional closed piecewise flat Alexandrov spaces

  • Xueping LiEmail author
Research Article


We show that a closed piecewise flat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdroff topology, there is a Lipschitz Gromov-Hausdorff approximation.


Alexandrov space Gromov-Hausdorff approximation tubular neighborhood 


53C20 53C99 57N65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Part of this work was done during the author’s visit to Capital Normal University, thanks a lot for their hospitality. The author would also like to thank the referees for lots of useful comments, especially the one leads to the simplification of the proof of Lemma 1. The author was in great debt to Prof. Xiaochun Rong, without whose help this work can never get the present form. This work was partially supported by the National Natural Science Foundation of China (Grant No. 11501258) and a foundation from Jiangsu Normal University.


  1. 1.
    Alexander S, Kapovitch V, Petrunin A. Alexandrov Geometry. Book in preparation, 2017zbMATHGoogle Scholar
  2. 2.
    Burago D, Burago Yu, Ivanov S. A Course in Metric Geometry. Grad Stud Math, Vol 33. Providence: Amer Math Soc, 2001Google Scholar
  3. 3.
    Burago Yu, Gromov M, Perelman G. A. D. Alexandrov spaces with curvature bounded below. Uspekhi Mat Nauk, 1992, 47(2): 3–51 (in Russian); Russian Math Surveys, 1992, 47(2): 1–58MathSciNetGoogle Scholar
  4. 4.
    Cheeger J, Colding T. On the structure of space with Ricci curvature bounded below I. J Differential Geom, 1997, 46: 406–480MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheeger J, Fukaya K, Gromov M. Nilpotent structures and invariant metrics on collapsed manifolds. J Amer Math Soc, 1992, 5: 327–372MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fukaya K. Collapsing of Riemannian manifolds to ones of lower dimensions. J Differential Geom, 1987, 25: 139–156MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gromov M, Lafontaine J, Pansu P. Structures métriques pour les variétés riemanniennes. Paris: CedicFernand, 1981Google Scholar
  8. 8.
    Perelman G. Alexandrov’s spaces with curvature bounded from below II. Preprint, 1991Google Scholar
  9. 9.
    Yamaguchi T. Collapsing and pinching under lower curvature bound. Ann of Math, 1991, 133: 317–357MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yamaguchi T. A convergence theorem in the geometry of Alexandrov spaces. Sémin Congr, 1996, 1: 601–642MathSciNetzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJiangsu Normal UniversityXuzhouChina

Personalised recommendations