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

Abstract

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-Hausdorff topology, there is a Lipschitz Gromov-Hausdorff approxima-tion.

Key words： Alexandrov space; Gromov-Hausdorff approximation; tubular neighborhood

References

1 Alexander S, Kapovitch V, Petrunin A. Alexandrov Geometry. Book in preparation, 2017

2 Burago D, Burago Yu, Ivanov S. A Course in Metric Geometry. Grad Stud Math, Vol 33. Providence: Amer Math Soc, 2001

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): 1–58

4 Cheeger J, Colding T. On the structure of space with Ricci curvature bounded below I. J Differential Geom, 1997, 46: 406–480

5 Cheeger J, Fukaya K, Gromov M. Nilpotent structures and invariant metrics on collapsed manifolds. J Amer Math Soc, 1992, 5: 327–372

6 Fukaya K. Collapsing of Riemannian manifolds to ones of lower dimensions. J Differential Geom, 1987, 25: 139–156

7 Gromov M, Lafontaine J, Pansu P. Structures métriques pour les variétés riemanniennes. Paris: CedicFernand, 1981

8 Perelman G. Alexandrov's spaces with curvature bounded from below II. Preprint, 1991

9 Yamaguchi T. Collapsing and pinching under lower curvature bound. Ann of Math, 1991, 133: 317{357

10 Yamaguchi T. A convergence theorem in the geometry of Alexandrov spaces. Sémin Congr, 1996, 1: 601–642