Abstract
In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ℂn. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].
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References
Bando, S. On three-dimensional compact Kähler manifolds of nonnegative bisectional curvature.J. Differential Geometry,19, 283–297, (1984).
Berger, M. Sur les groupes d’holonomie homogénes des variétés a connexion affine et des variétés riemanniennes,Bull. Soc. Math. France,83, 279–330, (1955).
Bishop, R.L. and Crittenden, R.J.Geometry of Manifolds, Academic Press, (1964).
Calabi, E. The space of Kähler metrics,Proc. Internat. Congress Math. Amsterdam. 2, 206–207, (1954).
Calabi, E. Improper affine hypersphere and generalization of a theorem of K. Jörgens,Mich. Math. J.,5, 105–126, (1958).
Cao, H.D. Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds,Invent. Math.,81, 359–372, (1985).
Cheeger, J. and Ebin, D.Comparison Theorems in Riemannian Geometry, North-Holland Math. Library, North-Holland, Amsterdam, (1975).
Gromoll, D. and Meyer, W. On complete open manifolds of positive curvature,Ann. of Math.,90, 75–90, (1969).
Cheng, S.Y. and Yau, S.T. Differential equations on Riemannian manifolds and their geometric applications,Comm. Pure Appl. Math.,28, 333–354, (1975).
Croke, C. Some isoperimetric inequalities and consequences,Ann. Scient. Ec. Norm. Sup.,13, (1980).
De Turck, D.M. Deforming metrics in direction of their Ricci tensors,J. Differential Geometry,18, 157–162, (1983).
Donnelly, H. Bounded harmonic functions and positive Ricci curvature,Math. Z,191, 559–565, (1986).
Greene, R.E. and Wu, H.Analysis on Noncompact Kähler Manifolds, Proc. Symp. Pure Math.,30, AMS, (1977).
Greene, R.E. and Wu, H. C∞-convex functions and manifolds of positive curvature,Acta Math.,137, 209–245, (1976).
Greene, R.E. and Wu, H.Function Theory on Manifolds which Possess a Pole, Lecture Notes in Math.,669, Springer-Verlag, Berlin-Heidelberg-New York, (1979).
Howard, A., Smyth, B., and Wu, H. On compact Kähler manifolds of nonnegative bisectional curvature, I,Acta Math.,147, 51–56, (1981).
Hamilton, R.S. Three-manifolds with positive Ricci curvature,J. Differential Geometry,17, 255–306, (1982).
Hamilton, R.S. Four-manifolds with positive curvature operator,J. Differential Geometry,24, 153–179, (1986).
Huisken, G. Ricci deformation of the metric on a Riemannian manifold,J. Differential Geometry,21, 47–62, (1985).
Li, P. and Yau, S.T. On the parabolic kernel of the Schrödinger operator,Acta Math.,156, 153–201, (1986).
Mok, N., Siu, Y.T., and Yau, S.T. The Poincaré-Lelong equation on complete Kähler manifolds,Comp. Math.,44, 183–218,(1981).
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).
Mok, N. The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature,J. Differential Geometry,27, 179–214, (1988).
Moser, J. On Harnack’s theorem for elliptic differential equations,Comm. Pure and Appl. Math.,14, 577–591, (1961).
Protter, M.H. and Weinberger, H.F.Maximum Principle in Differential Equations, Prentice-Hall, NJ, (1967).
Shi, W.X. Deforming the metric on complete Riemannian manifolds,J. Differential Geometry,30, 223–301, (1989).
Shi, W.X. Ricci deformation of the metric on complete noncompact Riemannian manifolds,J. Differential Geometry,30, 303–394, (1989).
Shi, W.X. Complete noncompact three-manifolds with nonnegative Ricci curvature,J. Differential Geometry,29, 353–360, (1989).
Shi, W.X. Ricci deformation of the metric on complete noncompact Kähler manifolds, Ph.D. Thesis, Harvard University, (1990).
Shi, W.X. Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature,Bull. of the American Mathematical Society,23, 437–440, (1990).
Schoen, R. and Yau, S.T. Complete three dimensional manifolds with positive Ricci curvature and scalar curvature, InSeminar on Differential Geometry, 209–228, Princeton University Press, Princeton, NJ, (1982).
Simons, J. On the transitivity of holnomy systems,Ann. of Math.,76, 213–234, (1962).
Yau, S.T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I,Comm. Pure and Appl. Math.,31, 339–411, (1978).
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Shi, WX. A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature. J Geom Anal 8, 117–142 (1998). https://doi.org/10.1007/BF02922111
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DOI: https://doi.org/10.1007/BF02922111