Soul theorem for 4dimensional topologically regular open nonnegatively curved Alexandrov spaces
Author:
Jian Ge
Journal:
Proc. Amer. Math. Soc. 139 (2011), 44354443
MSC (2010):
Primary 53C20
Published electronically:
April 13, 2011
MathSciNet review:
2823089
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Abstract: In this paper, we study the topology of topologically regular 4dimensional open nonnegatively curved Alexandrov spaces. These spaces occur naturally as the blowup limits of compact Riemannian manifolds with lower curvature bound. These manifolds have also been studied by Yamaguchi in his preprint. Our main tools are gradient flows of semiconcave functions and critical point theory for distance functions, which have been used to study the dimensional collapsing theory in the paper by Cao and Ge. The results of this paper will be used in our future studies of collapsing 4manifolds, which will be discussed elsewhere.
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 Jianguo Cao, Bo Dai, Jiaqiang Mei, An extension of Perelman's soul theorem for singular spaces, arXiv:0706.0565v5 [math.DG].
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 [CaoG10]
 Jianguo Cao, Jian Ge, A simple proof of Perelman's Collapsing Theorem for 3manifolds. The Journal of Geometric Analysis, published online August 5, 2010, DOI 10.1007/s1222001091695.
 [CG72]
 J. Cheeger, D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413443. MR 0309010 (46:8121)
 [FY92]
 K. Fukaya, T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Ann. of Math. (2) 136 (1992), 253333. MR 1185120 (93h:53041)
 [GP89]
 K. Grove, P. Petersen, On the excess of metric spaces and manifolds, preprint.
 [GM69]
 D. Gromoll, W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 7590. MR 0247590 (40:854)
 [Kap02]
 V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal. 12 (2002), no. 1, 121137. MR 1904560 (2003b:53043)
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 V. Kapovitch, Perelman's Stability Theorem, in ``Surveys in differential geometry'', Vol. XI. Metric and comparison geometry. Edited by Jeff Cheeger and Karsten Grove. International Press, Somerville, MA, 2007, pp. 103136. MR 2408265 (2009g:53057)
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 G. Perelman, A. D. Alexandrov's spaces with curvatures bounded from below. II, preprint.
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 G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. 40 (1994), 209212. MR 1285534 (95d:53037)
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 A. Petrunin, Semiconcave functions in Alexandrov's geometry. Surveys in differential geometry. Vol. XI. Metric and comparison geometry. Edited by Jeff Cheeger and Karsten Grove. Surveys in Differential Geometry, volume 11. International Press, Somerville, MA, 2007, pp. 137201. MR 2408266 (2010a:53052)
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 T. Shioya, T. Yamaguchi, Collapsing threemanifolds under a lower curvature bound, J. Differential Geom. 56 (2000), 166. MR 1863020 (2002k:53074)
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 H. Wu, An elementary method in the study of nonnegative curvature, Acta Math. 142 (1979), no. 12, 5778. MR 512212 (80c:53054)
 [Yam02]
 T. Yamaguchi, Collapsing 4manifolds under a lower curvature bound, 2002 preprint.
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Additional Information
Jian Ge
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email:
jge@nd.edu
DOI:
http://dx.doi.org/10.1090/S000299392011108311
Received by editor(s):
September 27, 2010
Received by editor(s) in revised form:
October 8, 2010
Published electronically:
April 13, 2011
Communicated by:
Jianguo Cao
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
