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Collapsing three-dimensional closed Alexandrov spaces with a lower curvature bound

Authors: Ayato Mitsuishi and Takao Yamaguchi
Journal: Trans. Amer. Math. Soc. 367 (2015), 2339-2410
MSC (2010): Primary 53C20, 53C23
Published electronically: November 24, 2014
MathSciNet review: 3301867
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Abstract: In the present paper, we determine the topologies of three-dimensional closed Alexandrov spaces which converge to lower dimensional spaces in the Gromov-Hausdorff topology.

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  • [BBI] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418 (2002e:53053)
  • [BGP] Yu. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3-51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1-58. MR 1185284 (93m:53035),
  • [BH] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • [CaGe] Jianguo Cao and Jian Ge, A simple proof of Perelman's collapsing theorem for 3-manifolds, J. Geom. Anal. 21 (2011), no. 4, 807-869. MR 2836584,
  • [ChGr] Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413-443. MR 0309010 (46 #8121)
  • [E] D. B. A. Epstein, Curves on $ 2$-manifolds and isotopies, Acta Math. 115 (1966), 83-107. MR 0214087 (35 #4938)
  • [FY] Kenji Fukaya and Takao Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2) 136 (1992), no. 2, 253-333. MR 1185120 (93h:53041),
  • [GP] Karsten Grove and Peter Petersen, A radius sphere theorem, Invent. Math. 112 (1993), no. 3, 577-583. MR 1218324 (94e:53034),
  • [Kap Rest] Vitali Kapovitch, Restrictions on collapsing with a lower sectional curvature bound, Math. Z. 249 (2005), no. 3, 519-539. MR 2121738 (2005k:53048),
  • [Kap Stab] Vitali Kapovitch, Perelman's stability theorem, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 103-136. MR 2408265 (2009g:53057),
  • [KL] B. Kleiner and J. Lott, Locally collapsed $ 3$-manifolds, preprint.
  • [KL orb] B. Kleiner and J. Lott, Geometrization of three-dimensional orbifolds via Ricci flow, preprint.
  • [KMS] Kazuhiro Kuwae, Yoshiroh Machigashira, and Takashi Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269-316. MR 1865418 (2002m:58052),
  • [L] A. Lytchak, Differentiation in metric spaces, Algebra i Analiz 16 (2004), no. 6, 128-161; English transl., St. Petersburg Math. J. 16 (2005), no. 6, 1017-1041. MR 2117451 (2005j:53039),
  • [Milka] A. D. Milka, Metric structure of a certain class of spaces that contain straight lines, Ukrain. Geometr. Sb. Vyp. 4 (1967), 43-48 (Russian). MR 0256327 (41 #983)
  • [M] Ayato Mitsuishi, A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications, Geom. Dedicata 144 (2010), 101-114. MR 2580420 (2011a:53062),
  • [MT] J. Morgan and G. Tian, Completion of the proof of the geometrization conjecture, preprint.
  • [N] M. A. Natsheh, On the involutions of closed surfaces, Arch. Math. (Basel) 46 (1986), no. 2, 179-183. MR 834833 (88a:57072),
  • [O] Yukio Otsu, On manifolds of small excess, Amer. J. Math. 115 (1993), no. 6, 1229-1280. MR 1254733 (95i:53046),
  • [OS] Yukio Otsu and Takashi Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994), no. 3, 629-658. MR 1274133 (95e:53062)
  • [Per II] G. Perelman, A. D. Alexandrov's spaces with curvatures bounded from below. II, preprint.
  • [Per Elem] G. Perelman, Elements of Morse theory on Aleksandrov spaces, Algebra i Analiz 5 (1993), no. 1, 232-241 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 205-213. MR 1220498 (94h:53054)
  • [Per DC] G. Perelman, DC Structure on Alexandrov Space, preprint.
  • [Per Ent] G.  Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG / 0211159.
  • [Per Surg] G.  Perelman, Ricci flow with surgery on three-manifolds, arXiv:math.DG / 0303109.
  • [PP QG] G. Perelman and A. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint.
  • [Pet QG] Anton Petrunin, Quasigeodesics in multidimensional Alexandrov spaces, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-University of Illinois at Urbana-Champaign, 1995. MR 2693118
  • [Pet Appl] Anton Petrunin, Applications of quasigeodesics and gradient curves, Comparison geometry (Berkeley, CA, 1993-94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203-219. MR 1452875 (98m:53061)
  • [Pet Para] Anton Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998), no. 1, 123-148. MR 1601854 (98j:53048),
  • [Pet Semi] Anton Petrunin, Semiconcave functions in Alexandrov's geometry, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137-201. MR 2408266 (2010a:53052),
  • [Pl] Conrad Plaut, Spaces of Wald curvature bounded below, J. Geom. Anal. 6 (1996), no. 1, 113-134. MR 1402389 (97j:53043),
  • [SY00] Takashi Shioya and Takao Yamaguchi, Collapsing three-manifolds under a lower curvature bound, J. Differential Geom. 56 (2000), no. 1, 1-66. MR 1863020 (2002k:53074)
  • [SY05] Takashi Shioya and Takao Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound, Math. Ann. 333 (2005), no. 1, 131-155. MR 2169831 (2006j:53050),
  • [Sie] L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123-136; ibid. 47 (1972), 137-163. MR 0319207 (47 #7752)
  • [W] Joseph A. Wolf, Spaces of constant curvature, 5th ed., Publish or Perish Inc., Houston, TX, 1984. MR 928600 (88k:53002)
  • [Y91] Takao Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), no. 2, 317-357. MR 1097241 (92b:53067),
  • [Y conv] Takao Yamaguchi, A convergence theorem in the geometry of Alexandrov spaces, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 601-642 (English, with English and French summaries). MR 1427772 (97m:53078)
  • [Y 4-dim] Takao Yamaguchi, Collapsing $ 4$-manifolds under a lower curvature bound, arXiv:1205.0323v1 [math.DG]
  • [Y ess] Takao Yamaguchi, Collapsing and essential covering, arXiv:1205.0441v1 [math.DG]

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Additional Information

Ayato Mitsuishi
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Takao Yamaguchi
Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan
Address at time of publication: Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Received by editor(s): June 29, 2012
Published electronically: November 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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