Soul theorem for 4-dimensional topologically regular open nonnegatively curved Alexandrov spaces

Author:
Jian Ge

Journal:
Proc. Amer. Math. Soc. **139** (2011), 4435-4443

MSC (2010):
Primary 53C20

DOI:
https://doi.org/10.1090/S0002-9939-2011-10831-1

Published electronically:
April 13, 2011

MathSciNet review:
2823089

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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 -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.

**[BBI01]**Dima Burago, Yuri Burago, S. Ivanov,*A course in metric geometry*. Graduate Studies in Mathematics,**33**. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. MR**1835418 (2002e:53053)****[BGP92]**Yu Burago, M. Gromov, G. Perelman,*A. D. Aleksandrov spaces with curvatures bounded below.*(Russian) Uspekhi Mat. Nauk**47**(1992), no. 2(284), 3-51, 222; translation in Russian Math. Surveys**47**(1992), no. 2, 1-58. MR**1185284 (93m:53035)****[CDM07]**Jianguo Cao, Bo Dai, Jiaqiang Mei,*An extension of Perelman's soul theorem for singular spaces*, arXiv:0706.0565v5 [math.DG].**[CDM09]**J. Cao, Bo Dai, Jiaqiang Mei,*An optimal extension of Perelman's comparison theorem for quadrangles and its applications.*Advanced Lectures in Mathematics, volume 11, pp. 39-59. In ``Recent Advances in Geometric Analysis'', edited by Y. Lee, C.-S. Lin, and M.-P. Tsui. Higher Educational Press and International Press, 2009. MR**2648938****[CaoG10]**Jianguo Cao, Jian Ge,*A simple proof of Perelman's Collapsing Theorem for 3-manifolds.*The Journal of Geometric Analysis, published online August 5, 2010, DOI 10.1007/s12220-010-9169-5.**[CG72]**J. Cheeger, D. Gromoll,*On the structure of complete manifolds of nonnegative curvature*, Ann. of Math. (2)**96**(1972), 413-443. MR**0309010 (46:8121)****[FY92]**K. Fukaya, T. Yamaguchi,*The fundamental groups of almost nonnegatively curved manifolds*, Ann. of Math. (2)**136**(1992), 253-333. 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), 75-90. MR**0247590 (40:854)****[Kap02]**V. Kapovitch,*Regularity of limits of noncollapsing sequences of manifolds*, Geom. Funct. Anal.**12**(2002), no. 1, 121-137. MR**1904560 (2003b:53043)****[Kap07]**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. 103-136. MR**2408265 (2009g:53057)****[Per91]**G. Perelman,*A. D. Alexandrov's spaces with curvatures bounded from below. II*, preprint.**[Per93]**G. Perelman,*Elements of Morse theory on Aleksandrov spaces*(Russian summary), Algebra i Analiz**5**(1993), no. 1, 232-241; translation in St. Petersburg Math. J.**5**(1994), no. 1, 205-213. MR**1220498 (94h:53054)****[Per94]**G. Perelman,*Proof of the soul conjecture of Cheeger and Gromoll*, J. Differential Geom.**40**(1994), 209-212. MR**1285534 (95d:53037)****[Petr07]**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. 137-201. MR**2408266 (2010a:53052)****[SY00]**T. Shioya, T. Yamaguchi,*Collapsing three-manifolds under a lower curvature bound*, J. Differential Geom.**56**(2000), 1-66. MR**1863020 (2002k:53074)****[Wu79]**H. Wu,*An elementary method in the study of nonnegative curvature*, Acta Math.**142**(1979), no. 1-2, 57-78. MR**512212 (80c:53054)****[Yam02]**T. Yamaguchi,*Collapsing 4-manifolds 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:
https://doi.org/10.1090/S0002-9939-2011-10831-1

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.