For more information about this meeting, contact Yakov Pesin, Anatole Katok, Svetlana Katok, Dmitri Burago.

Title: | Compactness for manifolds with bounded volume and diameter (THIS TALK WILL TAKE PLACE IN THE SYNERGISTIC ROOM) |

Seminar: | Center for Dynamics and Geometry Seminars |

Speaker: | Stefan Wenger, University of Illinois at Chicago |

Abstract: |

Gromov's compactness theorem for metric spaces asserts that every
uniformly compact sequence of metric spaces has a subsequence which converges in the Gromov-Hausdorff sense to a compact metric space. This
theorem has been of great importance in Riemannian an metric geometry, but
also other fields. I will show in this talk that if one replaces the
Hausdorff distance appearing in Gromov's theorem by the filling volume or
flat distance then every sequence of oriented k-dimensional Riemannian
manifolds with a uniform bound on diameter and volume has a subsequence
which converges in this new distance to a countably k-rectifiable metric
space. In general, such a sequence does not meet the conditions of
Gromov's theorem and does not have a subsequence which converges with
respect to the Gromov-Hausdorff distance.
The new distance mentioned above was first introduced and studied by
Christina Sormani and myself. In the talk, which will be self-contained, I
will explain the basic properties of this distance and illustrate it by
examples.
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### Room Reservation Information

Room Number: | MB106 |

Date: | 01 / 26 / 2009 |

Time: | 04:00pm - 05:00pm |