The tetrahedral property and a new Gromov–Hausdorff compactness theorem

Christina Sormani

doi:10.1016/j.crma.2013.02.011

Differential Geometry

The tetrahedral property and a new Gromov–Hausdorff compactness theorem
[La propriété tétraédrique et un nouvel théorème de compacité Gromov–Hausdorff]

Présenté par : the Editorial Board

Christina Sormani ¹

¹ CUNY Graduate Center and Lehman College, 365 Fifth Avenue, New York, NY 10016, USA

Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 119-122.

Résumé
Abstract

Nous présentons le théorème tétraédrique de compacité, qui stipule que les séquences de variétés riemanniennes avec une borne supérieure uniforme sur le volume et sur le diamètre, qui satisfont une propriété tétraédrique uniforme, admettent une sous-suite qui converge, au sens de Gromov–Hausdorff, vers un espace métrique dénombrable $H^{m}$ , rectifiable, de la même dimension. La propriété tétraédrique ne dépend que de la distance entre les points dans les sphères, mais nous montrons quʼelle fournit une borne inférieure sur le volume des boules.

We present the Tetrahedral Compactness Theorem, which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov–Hausdorff sense to a countably $H^{m}$ rectifiable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres; yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.

Reçu le : 2012-10-17
Accepté le : 2013-02-20
Publié le : 2013-02-01

DOI : 10.1016/j.crma.2013.02.011

Affiliations des auteurs :

Christina Sormani ¹

¹ CUNY Graduate Center and Lehman College, 365 Fifth Avenue, New York, NY 10016, USA

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     author = {Christina Sormani},
     title = {The tetrahedral property and a new {Gromov{\textendash}Hausdorff} compactness theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {119--122},
     publisher = {Elsevier},
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Christina Sormani. The tetrahedral property and a new Gromov–Hausdorff compactness theorem. Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 119-122. doi : 10.1016/j.crma.2013.02.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.02.011/

Bibliographie
Cité par

[1] Luigi Ambrosio; Bernd Kirchheim Currents in metric spaces, Acta Math., Volume 185 (2000) no. 1, pp. 1-80

[2] Mikhael Gromov Filling Riemannian manifolds, J. Differential Geom., Volume 18 (1983) no. 1, pp. 1-147

[3] Misha Gromov Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston Inc., Boston, MA, 1999 (based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates)

[4] Christina Sormani Properties of the intrinsic flat distance | arXiv

[5] Christina Sormani; Stefan Wenger Weak convergence and cancellation. Appendix by Raanan Schul and Stefan Wenger, Calc. Var. Partial Differential Equations, Volume 38 (2010) no. 1–2

[6] Christina Sormani; Stefan Wenger Intrinsic flat convergence of manifolds and other integral current spaces. Appendix by Christina Sormani, J. Differential Geom., Volume 87 (2011)

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