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 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.
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 , 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.
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Christina Sormani 1
@article{CRMATH_2013__351_3-4_119_0, 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}, volume = {351}, number = {3-4}, year = {2013}, doi = {10.1016/j.crma.2013.02.011}, language = {en}, }
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/
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