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Mathematische Annalen

, Volume 327, Issue 1, pp 1–23 | Cite as

On complete noncompact Kähler manifolds with positive bisectional curvature

  • Bing-Long Chen
  • Xi-Ping Zhu
Article

Abstract.

We prove that a complete noncompact Kähler manifold M n of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of C n and we show that the manifold is topologically R 2 n . In particular, when M n is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to C 2.

Keywords

Manifold Ricci Curvature Ahler Manifold Pseudoconvex Domain Geodesic Ball 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Cao, H.D.: On Harnack’s inequalities for the Kähler-Ricci flow. Invent. Math. 109, 247–263 (1992)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96, 413–443 (1972)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernal estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifold. J. Diff. Geom. 17, 15–53 (1982)MATHMathSciNetGoogle Scholar
  4. 4.
    Freedman, M.H.: The topology of four-dimensional manifolds. J. Diff. Geom. 17, 357–453 (1982)MATHGoogle Scholar
  5. 5.
    Grauert, H.: On Levi’s problem and the embedding of real-analytic manifolds. Ann. Math. 68, 460–472 (1958)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Greene, R.E., Wu, H.: C convex functions and manifolds of positive curvature. Acta Math. 137, 209–245 (1976)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gromoll, D., Meyer, W.: On complete open manifolds of positive curvature. Ann. Math. 90, 75–90 (1969)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Hamilton, R.S.: Formation of singularities in the Ricci flow. surveys in Diff. Geom. Vol. 2, Boston: International Press, 1995, pp. 7–136.Google Scholar
  9. 9.
    Hamilton, R.S.: A compactness property for solutions of the Ricci flow. Amer. J. Math. 117, 545–572 (1995)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Princeton, N.J., Van Nostrand, 1966Google Scholar
  11. 11.
    Markoe, A.: Runge families and increasing unions of Sein spaces. Bull. Amer. Math. Soc. 82, 787–788 (1976)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Mok, N.: The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Diff. Geom. 27, 179–214 (1988)MATHMathSciNetGoogle Scholar
  13. 13.
    Mok, N.: An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties. Bull. Soc. Math. France. 112, 197–258 (1984)MATHMathSciNetGoogle Scholar
  14. 14.
    Mok, N.: An embedding theorem of complex Kähler manifolds of positive Ricci curvature onto quasi-projective varieties. Math. Ann. 286, 373–408 (1990)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mok, N., Siu, Y.T., Yau, S.T.: The Poincaré-Lelong equation on complete Kähler manifolds. Comp. Math. 4, 183–218 (1981)MathSciNetGoogle Scholar
  16. 16.
    Ni, L.: Vanishing theorems on complete Kähler manifolds and their applications. J. Diff. Geom. 50, 89–122 (1998)MATHGoogle Scholar
  17. 17.
    Ramanujam, C.D.: A topological characterization of the affine plane as an algebraic variety. Ann. Math. 94, 69–88 (1971)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Schoen, R., Yau, S.T.: Lectures on differential geometry, in conference proceedings and Lecture Notes in Geometry and Topology. Vol. 1, International Press Publications, 1994Google Scholar
  19. 19.
    Shi, W.X.: Deforming the metric on complete Riemannian manifolds J. Diff. Geom. 30, 223–301 (1989)MATHGoogle Scholar
  20. 20.
    Shi, W.X.: Complete noncompact Kähler manifolds with positive bisectional curvature. Bull. Amer. Math. Soc. 23, 437–440 (1990)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Shi, W.X.: Ricci deformation of the metric on complete noncompact Kähler manifolds. Ph.D. Thesis: Harvard University, 1990Google Scholar
  22. 22.
    Shi, W.X.: Ricci flow and the uniformization on complete noncompact Kähler manifolds. J. Diff. Geom. 45, 94–220 (1997)MATHGoogle Scholar
  23. 23.
    Siu, Y.T.: Pseudoconvexity and the problem of Levi. Bull. Amer. Math. Soc. 84, 481–512 (1978)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Smale, S.: Generalised Poincaré’s conjecture in dimension greater than four. Ann. Math. 74, 391–466 (1961)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    To W.-K.: Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci Curvature and satisfying certain topological condictions. Duke Math. J. 63, 745–789 (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Bing-Long Chen
    • 1
  • Xi-Ping Zhu
    • 1
  1. 1.Department of MathematicsZhongshan UniversityGuangzhouP. R. China

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