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Taming 3-manifolds using scalar curvature

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Abstract

In this paper we address the issue of uniformly positive scalar curvature on noncompact 3-manifolds. In particular we show that the Whitehead manifold lacks such a metric, and in fact that \({\mathbb{R}^3}\) is the only contractible noncompact 3-manifold with a metric of uniformly positive scalar curvature. We also describe contractible noncompact manifolds of higher dimension exhibiting this curvature phenomenon. Lastly we characterize all connected oriented 3-manifolds with finitely generated fundamental group allowing such a metric.

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References

  1. Andrist K., Wright D.: The Whitehead manifold has no orientation reversing homeomorphism. Top. Proc. 31(1), 1–5 (2007)

    MATH  MathSciNet  Google Scholar 

  2. Block J., Weinberger S.: Arithmetic manifolds of positive scalar curvature. J. Diff. Geom. 52(2), 375–406 (1999)

    MATH  MathSciNet  Google Scholar 

  3. Brown M.: A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc. 66, 74–76 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cao H., Zhu X.: A complete proof of the Poincaré and geometrization conjectures: application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10(2), 165–498 (2006)

    MATH  MathSciNet  Google Scholar 

  5. Chang S.: Coarse obstructions to positive scalar curvature metrics in noncompact arithmetic manifolds. J. Diff. Geom. 57(1), 1–22 (2001)

    MATH  Google Scholar 

  6. Christ, U., Lohkamp, J.: Singular minimal hypersurfaces and scalar curvature, preprint.

  7. Connes A., Gromov M., Moscovici H.: Group cohomology with Lipschitz control and higher signature. Geom. Funct. Anal. 3, 1–78 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gromov M., Lawson B.: The classification of simply connected manifolds of positive scalar curvature. Ann. Math. 111, 423–434 (1980)

    Article  MathSciNet  Google Scholar 

  9. Gromov M., Lawson B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. IHES 58, 295–408 (1983)

    Google Scholar 

  10. Kervaire M.: Smooth homology spheres and their fundamental groups. Trans. Am. Math. Soc. 144, 67–72 (1969)

    MATH  MathSciNet  Google Scholar 

  11. Kleiner B., Lott J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lohkamp J.: Positive scalar curvature in dim ≥8. C.R. Math. Acad. Sci. Paris 343(9), 585–588 (2006)

    MATH  MathSciNet  Google Scholar 

  13. Mathai V.: The Novikov conjecture for low degree cohomology classes. Geom. Dedicata. 99, 1–15 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Matthey M., Oyono-Oyono H., Pitsch W.: Homotopy invariance of higher signatures and 3-manifold groups. Bull. Soc. Math. France. 136(1), 1–25 (2008)

    MATH  MathSciNet  Google Scholar 

  15. Mazur B.: A note on some contractible 4-manifolds. Ann. Math. 73(2), 221–228 (1961)

    Article  MathSciNet  Google Scholar 

  16. Morgan J., Tian, G.: Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3. Am. Math. Soc., Providence, RI; Clay Mathematics Institute, Cambridge, MA (2007)

  17. Nabutovsky A., Weinberger S.: Variational problems for Riemannian functionals and arithmetic groups. Publ. d’IHES 92, 5–62 (2000)

    MATH  MathSciNet  Google Scholar 

  18. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. (2002). arXiv:math/0211159

  19. Perelman, G.: Ricci flow with surgery on three-manifolds. (2003). arXiv:math/0303109

  20. Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. (2003). arXiv:math/0307245

  21. Pimsner M.: KK-groups of crossed products by groups acting on trees. Invent. Math. 86(3), 603–634 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  22. Roe J.: Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Amer. Math. Soc. 104, 497 (1993)

    MathSciNet  Google Scholar 

  23. Rosenberg J.: C *-algebras, positive scalar curvature, and the Novikov conjecture. Inst. Hautes Études Sci. Publ. Math. 58, 197–212 (1983)

    Article  Google Scholar 

  24. Rosenberg, J.: C*-algebras, positive scalar curvature and the Novikov conjecture II, Geometric methods in operator algebras (Kyoto, 1983), 341–374, Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow (1986)

  25. Rosenberg J.: C*-algebras, positive scalar curvature, and the Novikov conjecture III. Topology 25(3), 319–336 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schoen R., Yau S.-T.: On the structure of manifolds with positive scalar curvature. Manuscripta. Math. 28(1–3), 159–183 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  27. Skandalis G., Tu J.L., Yu G.: The coarse Baum–Connes conjecture and groupoids. Topology 41, 807–834 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Yu G.: The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139(1), 201–240 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Waldhausen F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87(2), 56–88 (1968)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Wellesley College, Wellesley, MA, 02481, USA

    Stanley Chang

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Shmuel Weinberger

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Guoliang Yu

Authors
  1. Stanley Chang
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  2. Shmuel Weinberger
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  3. Guoliang Yu
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Corresponding author

Correspondence to Shmuel Weinberger.

Additional information

Research partially supported by NSF Grants DMS-0405867, DMS-0805913 and DMS-0600216.

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Chang, S., Weinberger, S. & Yu, G. Taming 3-manifolds using scalar curvature. Geom Dedicata 148, 3–14 (2010). https://doi.org/10.1007/s10711-009-9402-1

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  • DOI: https://doi.org/10.1007/s10711-009-9402-1

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