Abstract
We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.
Highlights
We review integral currents on metric spaces as introduced by Ambrosio and Kirchheim [1], the intrinsic flat distance introduced by Sormani and Wenger [13] and some theory on the structure of spaces with Ricci curvature bounded below by Cheeger and Colding [3,4]
We denote the metric space MnI F /ι endowed with this quotient metric by MnI F/ι
In [3,4,5], Cheeger and Colding study the structure of metric spaces that arise as the Gromov-Hausdorff limits of manifolds with Ricci curvature uniformly bounded from below
Summary
We review integral currents on metric spaces as introduced by Ambrosio and Kirchheim [1], the intrinsic flat distance introduced by Sormani and Wenger [13] and some theory on the structure of spaces with Ricci curvature bounded below by Cheeger and Colding [3,4]. The prime purpose of this review is to fix notation. We adhere closely to the notation used in these articles, and the reader familiar with these works could probably understand the rest of the manuscript without reading this section
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