Compactness for manifolds and integral currents with bounded diameter and volume

Details Compactness for manifolds and integral currents with bounded diameter and volume Compactness for manifolds and integral currents with bounded diameter and volume

2008-09-18

arxiv.org/abs/0809.3257v2

Mathematics

Differential Geometry

Some changes made to the introduction; some references added

Scientific paper

By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented $k$-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio-Kirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which we replace uniform compactness of the sequence with uniform bounds on volume and diameter.

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