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Universal Covers for Hausdor Limits of Noncompact Spaces Christina Sormani Guofang Wei y

Summary: Universal Covers for Hausdor Limits of Noncompact Spaces
Christina Sormani  Guofang Wei y
We prove that if Y is the Gromov-Hausdor limit of a sequence of complete manifolds, M n
i ,
with a uniform lower bound on Ricci curvature then Y has a universal cover.
1 Introduction
One of the main trends in Riemannian Geometry today is the study of Gromov Hausdor limits. The
starting point is Gromov's precompactness theorem. Namely a sequence of complete Riemannian
manifolds with a uniform lower bound on their Ricci curvature have a converging subsequence.
Moreover the limit space is a complete length space. That is, it is a metric space such that between
every two points there is a length minimizing curve whose length is the distance between the two
points. See [Gr], also [BBI]. To prove this theorem, the only property of Ricci curvature that Gromov
uses is the Bishop-Gromov volume comparison theorem [BiCr][Gr] which provides an estimate on
the number of disjoint small balls which t in a large ball.
When the sectional curvature of the sequence is uniformly bounded from below, the limit space
is much understood. Namely it is an Alexandrov space with curvature bounded below and by the
work of Perelman [Pl1] it is a strati ed topological manifold and is locally contractible.
In the case when only Ricci curvature is bounded from below Menguy has shown the limit space
can even have in nite topological type on arbitrarily small balls even with an additional assumption


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics