A uniformization theorem for complete non-compact kähler surfaces with positive bisectional curvature

Journal of Differential Geometry

Volumn 67, Issue 3, 2004, Pages 519-570

  Chen, Bing Long ^a   Tang, Siu Hung ^a,b,c   Zhu, Xi Ping ^a  

^a SUN YAT SEN UNIVERSITY   (China)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^c UNIVERSITY OF KENTUCKY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33744999185     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1102091357     Document Type: Article

References (41)

1. A. Andreotti & E. Vesentini, Carleman estimates for the Laplacian-Beltrami operator on complex manifolds, Publ. Math. Inst. Hantes Etudes Sci. 25 (1965) 313-362.
2. R.L. Bishop & S.I. Goldberg, Some implications of the generalized Gauss-Bennet theorem, Trans. Am. Math. Soc. 112 (1964) 508-535.
3. D. Burns, S. Shnider, & R.O. Wells, On deformations of strictly pseudoconvex domain, Invent. Math. 46 (1978) 237-253.
4. H.-D. Cao, On Harnack’s inequalities for the Kahler-Ricci flow, Invent. Math. 109 (1992) 247-263.
5. J. Cheeger, M. Gromov, & M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982) 15-53.
6. B.L. Chen, On the geometry of complete positively curved Kahler manifolds, thesis, The Chinese University of Hong Kong, 2003.
7. B.L. Chen & X.P. Zhu, Complete Riemannian manifolds with pointwise pinched curvature, Invent. Math. 140 (2000) 423-452.
8. B.L. Chen & X.P. Zhu, On complete non-compact Kahler manifolds with positive bisectional curvature, Math. Ann. 327(1) (2003) 1-23.
9. B.L. Chen & X.P. Zhu, Volume growth and curvature decay of positively curved Kahler manifolds, ArXiv: math.DG/0211374.
10. S.Y. Cheng & S.-T. Yau, On the existence of a complete Kahler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 (1980) 507-544.
11. C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974) 1-65.
12. T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961) 157-170.
13. M.H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357-453.
14. R.E. Greene & H. Wu, C“convex functions and manifolds of positive curvature, Acta. Math. 137 (1976) 209-245.
15. R.E. Greene & H. Wu, Analysis on non-compact Kahler manifolds, Proc. Symp. Pure. Math., 30, Part II, Am. Math. Soc., 1977, 69-100.
16. R.S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153-179.
17. R.S. Hamilton, A compactness property for solution of the Ricci flow, Am. J. Math. 117 (1995) 545-572.
18. R.S. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry 2 (1995) 7-136, International Press.
19. 2-estimates and existence theorems for the 8-operator, Acta. Math. 113 (1965) 89-152.
20. T. Ivey, Ricci solitons on compact Kahler surfaces, Proc. of AMS 125 (1997) 1203-1208.
21. 3 -actions and algebraic threefolds with ample tangent bundle, Nagoya Math. J. 69 (1978) 33-64.
22. A. Markoe, Runge families and increasing unions of Stein spaces, Bull. Am. Math. Soc. 82 (1976) 787-788.
23. N. Mok, Y.T. Siu, & S.-T. Yau, The Poincare-Lelong equation on complete Kahler manifolds, Compositio Math. 44 (1981) 183-218.
24. N. Mok, An embedding theorem of complete Kahler manifolds of positive bisectional curvature onto affine algebrari varieties, Bull. Soc. Math. France 112 (1984) 197-258.
25. N. Mok, The uniformization theorem for compact Kahler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988) 179-214.
26. N. Mok, An embedding theorem of complete Kahler manifolds of positive Ricci curvature onto quasi-projective varieties, Math. Ann. 286 (1990) 373-408.
27. S. Mori, Projective manifolds with ample tangent bundles, Ann. Math. 100 (1979) 593-606.
28. L. Ni, Y.G. Shi, & L.F. Tam, Poisson equation, Poincaré-Lelong equation and curvature decay on complete Kahler manifolds, J. Differential Geom. 57 (2001) 339-388.
29. L. Ni & L.F. Tam, Plurisubharmonic functions and the structure of complete Kahler manifolds with non-negative curvature, J. Differential Geom. 64 (2003) 457-524.
30. K. Oka, Domaines finis sans point critique intérieur, Jap. J. Math. 23 (1953) 97-155.
31. C.P. Ramanujam, A topological charaterization of the affine plane as an algebraic variety, Ann. Math. 94 (1971) 69-88.
32. R. Schoen & S.-T. Yau, Lectures on differential geometry, in ‘Conference proceedings and Lecture Notes in Geometry and Topology’, 1, International Press Publications, 1994.
33. W.X. Shi, Deforming the metric on complete Riemannian manifold, J. Differential Geom. 30 (1989) 223-301.
34. W.X. Shi, Complete non-compact Kahler manifolds with positive holomorphic bisectional curvature, Bull. Am. Math. Soc. 23 (1990) 437-440.
35. W.X. Shi, Ricci deformation of the metric on complete non-compact Kaahler manifolds, Ph.D. Thesis Harvard University, 1990.
36. W.X. Shi, Ricci flow and the uniformization on complete non-compact Kahler manifolds, J. Differential Geom. 45 (1997) 94-220.
37. Y.T. Siu, Pseudoconvexity and the problem of Levi, Bull. Am. Math. Soc. 84 (1978) 481-512.
38. Y.T. Siu & S.-T. Yau, Compact Kahler manifolds of positive bisectional curvature, Invent. Math. 59 (1980) 189-204.
39. W.K. To, Quasi-projective embeddings of non-compact complete Kahler manifolds of positive Ricci curvature and satisfying certain topological conditions, Duke Math. J. 63(3) (1991) 745-789.
40. S.-T. Yau, Problem Section, in Seminar on Differential Geomeyry (S.T. Yau ed.), Princeton University Press, 1982, 669-706.
41. S.-T. Yau, A review of complex differential geometry, Proc. Symp. Pure Math. 52, Part II, Am. Math. Soc. 1991, 619-625.