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arXiv · 1411.6854

On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces

Nan Li·Raquel Perales
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Abstract

We study sequences of integral current spaces $(X_j,d_j,T_j)$ such that the integral current structure $T_j$ has weight $1$ and no boundary and, all $(X_j,d_j)$ are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either their limits collapse or the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat limits agree. The latter is done showing that the lower $n$ dimensional density of the mass measure at any regular point of the Gromov-Hausdorff limit space is positive by passing to a filling volume estimate. In an appendix we show that the filling volume of the standard $n$ dimensional integral current space coming from an $n$ dimensional sphere of radius $r>0$ in Euclidean space equals $r^n$ times the filling volume of the $n$ dimensional integral current space coming from the $n$ dimensional sphere of radius $1$.

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Nan Li, Raquel Perales. 2014-11-25. On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces. https://arxiv.org/abs/1411.6854

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