On graph approximations of surfaces with small area
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N. Zinov′ev
Translated by: the author - St. Petersburg Math. J. 17 (2006), 435-442
- DOI: https://doi.org/10.1090/S1061-0022-06-00912-5
- Published electronically: March 9, 2006
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Abstract:
It is shown that, for every closed oriented surface $M$ of genus $g$ with an arbitrary Riemann metric, there exists a metric graph of genus at most $g$ such that the Gromov–Hausdorff distance between $M$ and $\Gamma$ does not exceed $C\sqrt {\operatorname{Vol}M}$, where $C$ depends only on $g$.References
- Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
- Ju. D. Burago and V. A. Zalgaller, Geometricheskie neravenstva, “Nauka” Leningrad. Otdel., Leningrad, 1980 (Russian). MR 602952
- Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
Bibliographic Information
- N. Zinov′ev
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Avenue 28, Staryĭ Peterhof, St. Petersburg 198904, Russia
- Email: nikita@nz5608.spb.edu
- Received by editor(s): July 5, 2004
- Published electronically: March 9, 2006
- Additional Notes: Partially supported by NSF (grant DMS-0412166) and by NS (grant no. 1914.2003.1)
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 435-442
- MSC (2000): Primary 53C23
- DOI: https://doi.org/10.1090/S1061-0022-06-00912-5
- MathSciNet review: 2167844