Four-Manifolds, Curvature Bounds, and Convex Geometry

  • Claude LeBrun
Part of the Progress in Mathematics book series (PM, volume 271)


Seiberg—Witten theory leads to a remarkable family of curvature estimates governing the Riemannian geometry of compact 4-manifolds. This chapter describes a more transparent and user-friendly repackaging of these estimates, formulated in terms of the convex hull of the set of monopole classes. New results are then obtained concerning the boundary cases of the reformulated curvature estimates.


Curvature Bound Ahler Manifold Einstein Metrics Positive Scalar Curvature Convex Geometry 


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  1. 1.
    N. Aronszajn, A. Krzywicki, and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), pp. 417–453.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.Google Scholar
  3. 3.
    C. Bär, On nodal sets for Dirac and Laplace operators, Comm. Math. Phys., 188 (1997), pp. 709–721.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. Bauer, A stable cohomotopy refinement of Seiberg-Witten invariants. II, Invent. Math., 155 (2004), pp. 21–40.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    S. Bauer and M. Furuta, A stable cohomotopy refinement of Seiberg-Witten invariants. I, Invent. Math., 155 (2004), pp. 1–19.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987.MATHGoogle Scholar
  7. 7.
    D. E. Blair, The “total scalar curvature” as a symplectic invariant and related results, in Proceedings of the 3rd Congress of Geometry (Thessaloniki, 1991), Thessaloniki, 1992, Aristotle Univ. Thessaloniki, pp. 79–83.Google Scholar
  8. 8.
    J.-P. Bourguignon, Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein, Invent. Math., 63 (1981), pp. 263–286.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    S. K. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, J. Differential Geom., 24 (1986), pp. 275–341.MathSciNetGoogle Scholar
  10. 10.
    T. C. Drăghici, On some 4-dimensional almost Kähler manifolds, Kodai Math. J., 18 (1995), pp. 156–168.CrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Friedman and J. Morgan, Algebraic surfaces and Seiberg-Witten invariants, J. Alg. Geom., 6 (1997), pp. 445–479.MATHMathSciNetGoogle Scholar
  12. 12.
    D. T. Gay and R. Kirby, Constructing symplectic forms on 4-manifolds which vanish on circles, Geom. Topol., 8 (2004), pp. 743–777 (electronic).CrossRefMathSciNetGoogle Scholar
  13. 13.
    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second ed., 1983.MATHGoogle Scholar
  14. 14.
    M. Gromov and H. B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. Math., 111 (1980), pp. 423–434.CrossRefGoogle Scholar
  15. 15.
    F. Hirzebruch and H. Hopf, Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Ann., 136 (1958), pp. 156–172.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    N. Hitchin, Harmonic spinors, Adv. Math., 14 (1974), pp. 1–55.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    M. Ishida and C. LeBrun, Curvature, connected sums, and Seiberg-Witten theory, Comm. Anal. Geom., 11 (2003), pp. 809–836.MATHMathSciNetGoogle Scholar
  18. 18.
    T. P. Killingback and E. G. Rees, Spinc structures on manifolds, Classical Quantum Gravity, 2 (1985), pp. 433–438.CrossRefMathSciNetGoogle Scholar
  19. 19.
    N. Koiso, A decomposition of the space M of Riemannian metrics on a manifold, Osaka J. Math., 16 (1979), pp. 423–429.MATHMathSciNetGoogle Scholar
  20. 20.
    P. B. Kronheimer, Minimal genus in S 1 ×M 3, Invent. Math., 135 (1999), pp. 45–61.CrossRefMathSciNetGoogle Scholar
  21. 21.
    P. B. Kronheimer and T. S. Mrowka, Recurrence relations and asymptotics for four-manifold invariants, Bull. Am. Math. Soc. (N.S.), 30 (1994), pp. 215–221.CrossRefGoogle Scholar
  22. 22.
    —— , Scalar curvature and the Thurston norm, Math. Res. Lett., 4 (1997), pp. 931–937.MathSciNetGoogle Scholar
  23. 23.
    H. B. Lawson and M. Michelsohn, Spin Geometry, Princeton University Press, Princeton, NJ, 1989.Google Scholar
  24. 24.
    C. LeBrun, Einstein metrics and Mostow rigidity, Math. Res. Lett., 2 (1995), pp. 1–8.MATHMathSciNetGoogle Scholar
  25. 25.
    —— , Polarized 4-manifolds, extremal Kähler metrics, and Seiberg-Witten theory, Math. Res. Lett., 2 (1995), pp. 653–662.MATHMathSciNetGoogle Scholar
  26. 26.
    —— , Four-manifolds without Einstein metrics, Math. Res. Lett., 3 (1996), pp. 133–147.MATHMathSciNetGoogle Scholar
  27. 27.
    ——, Ricci curvature, minimal volumes, and Seiberg-Witten theory, Inv. Math., 145 (2001), pp. 279–316.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    ——, Einstein metrics, four-manifolds, and differential topology, in Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), Surv. Differ. Geom., VIII, International Press of Boston, Somerville, MA, 2003, pp. 235–255.Google Scholar
  29. 29.
    —— , Einstein metrics, symplectic minimality, and pseudo-holomorphic curves, Ann. Global Anal. Geom., 28 (2005), pp. 157–177.MATHMathSciNetGoogle Scholar
  30. 30.
    A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris, 257 (1963), pp. 7–9.MATHMathSciNetGoogle Scholar
  31. 31.
    R. Penrose and W. Rindler, Spinors and space-time. Vol. 1, Cambridge University Press, Cambridge, 1984. Two-spinor calculus and relativistic fields.CrossRefGoogle Scholar
  32. 32.
    C. H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett., 1 (1994), pp. 809–822.MathSciNetGoogle Scholar
  33. 33.
    E. Witten, Monopoles and four-manifolds, Math. Res. Lett., 1 (1994), pp. 809–822.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media LLC 2009

Authors and Affiliations

  • Claude LeBrun
    • 1
  1. 1.SUNY Stony BrookStony BrookNew YorkUSA

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