Advertisement

Geometriae Dedicata

, Volume 148, Issue 1, pp 3–14 | Cite as

Taming 3-manifolds using scalar curvature

  • Stanley Chang
  • Shmuel Weinberger
  • Guoliang Yu
Original Paper

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.

Keywords

Positive scalar curvature Noncompact manifolds Whitehead manifold 

Mathematics Subject Classification (2000)

53C21 19K56 57N10 57M40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrist K., Wright D.: The Whitehead manifold has no orientation reversing homeomorphism. Top. Proc. 31(1), 1–5 (2007)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Block J., Weinberger S.: Arithmetic manifolds of positive scalar curvature. J. Diff. Geom. 52(2), 375–406 (1999)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Brown M.: A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc. 66, 74–76 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 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)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Chang S.: Coarse obstructions to positive scalar curvature metrics in noncompact arithmetic manifolds. J. Diff. Geom. 57(1), 1–22 (2001)zbMATHGoogle Scholar
  6. 6.
    Christ, U., Lohkamp, J.: Singular minimal hypersurfaces and scalar curvature, preprint.Google Scholar
  7. 7.
    Connes A., Gromov M., Moscovici H.: Group cohomology with Lipschitz control and higher signature. Geom. Funct. Anal. 3, 1–78 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gromov M., Lawson B.: The classification of simply connected manifolds of positive scalar curvature. Ann. Math. 111, 423–434 (1980)CrossRefMathSciNetGoogle Scholar
  9. 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. 10.
    Kervaire M.: Smooth homology spheres and their fundamental groups. Trans. Am. Math. Soc. 144, 67–72 (1969)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Kleiner B., Lott J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lohkamp J.: Positive scalar curvature in dim ≥8. C.R. Math. Acad. Sci. Paris 343(9), 585–588 (2006)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Mathai V.: The Novikov conjecture for low degree cohomology classes. Geom. Dedicata. 99, 1–15 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 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)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Mazur B.: A note on some contractible 4-manifolds. Ann. Math. 73(2), 221–228 (1961)CrossRefMathSciNetGoogle Scholar
  16. 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)Google Scholar
  17. 17.
    Nabutovsky A., Weinberger S.: Variational problems for Riemannian functionals and arithmetic groups. Publ. d’IHES 92, 5–62 (2000)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. (2002). arXiv:math/0211159Google Scholar
  19. 19.
    Perelman, G.: Ricci flow with surgery on three-manifolds. (2003). arXiv:math/0303109Google Scholar
  20. 20.
    Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. (2003). arXiv:math/0307245Google Scholar
  21. 21.
    Pimsner M.: KK-groups of crossed products by groups acting on trees. Invent. Math. 86(3), 603–634 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Roe J.: Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Amer. Math. Soc. 104, 497 (1993)MathSciNetGoogle Scholar
  23. 23.
    Rosenberg J.: C *-algebras, positive scalar curvature, and the Novikov conjecture. Inst. Hautes Études Sci. Publ. Math. 58, 197–212 (1983)CrossRefGoogle Scholar
  24. 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)Google Scholar
  25. 25.
    Rosenberg J.: C*-algebras, positive scalar curvature, and the Novikov conjecture III. Topology 25(3), 319–336 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Schoen R., Yau S.-T.: On the structure of manifolds with positive scalar curvature. Manuscripta. Math. 28(1–3), 159–183 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Skandalis G., Tu J.L., Yu G.: The coarse Baum–Connes conjecture and groupoids. Topology 41, 807–834 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Waldhausen F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87(2), 56–88 (1968)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsWellesley CollegeWellesleyUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations