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$ 3$-manifolds with positive flat conformal structure


Authors: Reiko Aiyama and Kazuo Akutagawa
Journal: Proc. Amer. Math. Soc. 140 (2012), 3587-3592
MSC (2010): Primary 53A30, 53C21; Secondary 57M10
DOI: https://doi.org/10.1090/S0002-9939-2012-11423-6
Published electronically: February 15, 2012
MathSciNet review: 2929027
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Abstract: In this paper, we consider a closed $ 3$-manifold $ M$ with flat conformal structure $ C$. We will prove that if the Yamabe constant of $ (M, C)$ is positive, then $ (M, C)$ is Kleinian.


References [Enhancements On Off] (What's this?)

  • 1. K. Akutagawa, Aubin's lemma for the Yamabe constants of infinite coverings and a positive mass theorem, to appear in Math. Ann.
  • 2. T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296. MR 0431287 (55:4288)
  • 3. T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer, 1998. MR 1636569 (99i:58001)
  • 4. R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661-693. MR 849427 (88b:58144)
  • 5. J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math. 30 (1975), 1-41. MR 0418147 (54:6189)
  • 6. K. S. Brown, Cohomology of Groups, Graduate Texts in Math. 87, Springer, 1982. MR 672956 (83k:20002)
  • 7. M. Gromov and H. B. Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83-196. MR 720933 (85g:58082)
  • 8. M. Gromov, H. B. Lawson Jr. and W. Thurston, Hyperbolic $ 4$-manifolds and conformally flat $ 3$-manifolds, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 27-45. MR 1001446 (90k:57021)
  • 9. J. Hempel, $ 3$-Manifolds, Ann. of Math. Studies 86, Princeton Univ. Press, 1976. MR 0415619 (54:3702)
  • 10. H. Izeki, On the decomposition of conformally flat manifolds, J. Math. Soc. Japan 45 (1993), 105-119. MR 1195686 (93h:53036)
  • 11. H. Izeki, A deformation of flat conformal structures, Trans. Amer. Math. Soc. 348 (1996), 4939-4964. MR 1348862 (97c:53020)
  • 12. H. Izeki, Quasiconformal stability of Kleinian groups and an embedding of a space of flat conformal structures, Conform. Geom. Dyn. 4 (2000), 108-119. MR 1799652 (2002a:57056)
  • 13. O. Kobayashi, The scalar curvature of a metric with unit volume, Math. Ann. 279 (1987), 253-265. MR 919505 (89a:53048)
  • 14. N. H. Kuiper, On conformally flat spaces in the large, Ann. of Math. (2) 50 (1949), 916-924. MR 0031310 (11:133b)
  • 15. J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), 37-81. MR 888880 (88f:53001)
  • 16. T. Parker and C. Taubes, On Witten's proof of the positive energy theorem, Comm. Math. Phys. 84 (1982), 223-238. MR 661134 (83m:83020)
  • 17. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature,
    J. Differential Geom. 20 (1984), 479-495. MR 788292 (86i:58137)
  • 18. R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations, Lect. Notes in Math., vol. 1365, Springer, 1989, pp. 121-154. MR 994021 (90g:58023)
  • 19. R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47-71. MR 931204 (89c:58139)
  • 20. N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 (1968), 265-274. MR 0240748 (39:2093)
  • 21. E. Witten, A simple proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381-402. MR 626707 (83e:83035)
  • 22. H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka J. Math. 12 (1960), 21-37. MR 0125546 (23:A2847)

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Additional Information

Reiko Aiyama
Affiliation: Department of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan
Email: aiyama@math.tsukuba.ac.jp

Kazuo Akutagawa
Affiliation: Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
Email: akutagawa@math.is.tohoku.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2012-11423-6
Keywords: Differential geometry, geometric topology
Received by editor(s): April 5, 2011
Published electronically: February 15, 2012
Additional Notes: The second author was supported in part by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 21540097.
Communicated by: Lei Ni
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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