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Volume growth and the topology of pointed Gromov-Hausdorff limits
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Munn, Michael (2010) Volume growth and the topology of pointed Gromov-Hausdorff limits. Differential Geometry and Its Applications, Vol.28 (No.5). pp. 532-542. doi:10.1016/j.difgeo.2010.04.004
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Official URL: http://dx.doi.org/10.1016/j.difgeo.2010.04.004
Abstract
In this paper we examine topological properties of pointed metric measure spaces (Y, p) that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds {(M-i(n), p(i))}(i=1)(infinity) with nonnegative Ricci curvature. Cheeger and Colding (1997) [7] showed that given such a sequence of Riemannian manifolds it is possible to define a measure nu on the limit space (Y, p). In the current work, we generalize previous results of the author to examine the relationship between the topology of (Y, p) and its volume growth. Namely, given constants alpha(k, n) which were computed in Munn (2010) [16] and based on earlier work of G. Perelman, we show that if lim(r ->infinity) nu(B-p(r))/omega(n)r(n) > alpha(k, n), then the kth homotopy group of (Y, p) is trivial. The constants alpha(k, n) are explicit and depend only on n, the dimension of the manifolds {(M-i(n), p(i))}, and k, the dimension of the homotopy in (Y, p). (C) 2010 Elsevier B.V. All rights reserved.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science > Mathematics | ||||
| Journal or Publication Title: | Differential Geometry and Its Applications | ||||
| Publisher: | Elsevier BV * North-Holland | ||||
| ISSN: | 0926-2245 | ||||
| Official Date: | October 2010 | ||||
| Dates: |
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| Volume: | Vol.28 | ||||
| Number: | No.5 | ||||
| Number of Pages: | 11 | ||||
| Page Range: | pp. 532-542 | ||||
| DOI: | 10.1016/j.difgeo.2010.04.004 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Funder: | NSF, PSC-CUNY | ||||
| Grant number: | OISE-0754379, 60079-39 40 |
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