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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Soul theorem for 4-dimensional topologically regular open nonnegatively curved Alexandrov spaces
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by Jian Ge
Proc. Amer. Math. Soc. 139 (2011), 4435-4443
DOI: https://doi.org/10.1090/S0002-9939-2011-10831-1
Published electronically: April 13, 2011
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Abstract:

In this paper, we study the topology of topologically regular 4-dimensional open nonnegatively curved Alexandrov spaces. These spaces occur naturally as the blow-up 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 semi-concave functions and critical point theory for distance functions, which have been used to study the $3$-dimensional collapsing theory in the paper by Cao and Ge. The results of this paper will be used in our future studies of collapsing 4-manifolds, which will be discussed elsewhere.
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Bibliographic Information
  • Jian Ge
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
  • Email: jge@nd.edu
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 4435-4443
  • MSC (2010): Primary 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10831-1
  • MathSciNet review: 2823089