Ricci flow and the uniformization on complete noncompact kähler manifolds

Journal of Differential Geometry

Volumn 45, Issue 1, 1997, Pages 94-220

  Shi, Wan Xiong ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030635298     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1214459756     Document Type: Article

References (48)

1. E. Andersen &L. Lempert, On the group of holomorphic automorphisms of C”, Invent. Math. 110 (1992) 371-388.
2. A. Andreotti &E. Vesentini, Carleman estimates for the Laplace-Beltrami equations on complex manifolds, Inst. Hautes Etudes Sci. Publ. Math. 25 (1965) 81-130.
3. S. Bando, On three-dimensional compact Kdhler manifolds of nonnegative bisectional curvature, J. Differential Geom. 19 (1984) 283-297.
4. M. Berger, Sur les groupes d’holonomie homogènes des variétés a connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955) 279-330.
5. R. L. Bishop &R. J. Crittenden, Geometry of Manifolds, Academic Press, New York, 1964.
6. S. Bochner &W. T. Martin, Several Complex Variables, Princeton University Press, Princeton, NJ, 1948.
7. E. Calabi, The space of Kdhler metrics, Proc. Internat. Congress Math., Amsterdam 2 (1954) 206-207.
8. E. Calabi, Improper affine hypersphere and generalization of a theorem of K. Jorgens, Mich. Math. J. 5 (1958) 105-126.
9. H. D. Cao, Deformation of Kdhler metrics to Kdhler-Einstein metrics on compact Kdhler manifolds, Invent. Math. 81 (1985) 359-372.
10. H. D. Cao, On Harnack’s inequalities for the Kdhler-Ricci flow, Invent. Math. 109 (1992) 247-263.
11. J. Cheeger &D. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Math. Library, North-Holland, Amsterdam, 1975.
12. S. Y. Cheng &S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975) 333-354.
13. C. Croke, Some isoperimetric inequalities and consequences, Ann. Sci. Ecole Norm. Sup. 13 (1980).
14. D. M. DeTurck, Deforming metrics in direction of their Ricci tensors, J. Differential Geom. 18 (1983) 157-162.
15. P. G. Dixon &J. Esterle, Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon, Bull. Amer. Math. Soc. 15 (1986) 127-187.
16. 1, Invent. Math. 112 (1993) 323-349.
17. L. W. Green, A theorem of E. Hopf, Michigan Math. J. 5 (1958) 31-34.
18. R.E. Greene &H. Wu, Analysis on noncompact Kdhler manifolds, Proc. Symp. Pure Math., Amer. Math. Soc. 30 (1977).
19. R.E. Greene &H. Wu, C°° -convex functions and manifolds of positive curvature, Acta Math. 137 (1976) 209-245.
20. R.E. Greene &H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math. Vol. 669, Springer, Berlin, 1979.
21. D. Gromoll &W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. 90 (1969) 75-90.
22. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306.
23. R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153-179.
24. R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993) 225-243.
25. L. Hormander, An Introduction to Complex Analysis in Several Variables, D. Van Nostrand, Princeton, New Jersey, 1966.
26. 2 estimate and existence theorems for the d operator, Acta Math. 113 (1965) 89-152.
27. A. Howard, B. Smyth &H. Wu, On compact Kdhler manifolds of nonnegative bisectional curvature. I, Acta Math. 147 (1981) 51-56.
28. G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985) 47-62.
29. J. J. Kohn &L. Nirenberg, Non-coercive boundary problems, Comm. Pure Appl. Math. 18 (1965) 443-492.
30. P. Li &S. T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986) 153-201.
31. N. Mok, Y. T. Siu &S. T. Yau, The Poincare-Lelong eguation on complete Kdhler manifolds, Compositio Math. 44 (1981) 183-218.
32. N. Mok, An embedding theorem of complete Kdhler manifolds of positive bisectional curvature onto affine algebraic varieties, Bull. Soc. Math. France 112 (1984) 197-258.
33. N. Mok, The uniformization theorem for compact Kdhler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988) 179-214.
34. S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. 110 (1979) 593-606.
35. J. Moser, On Harnack’s theorem for elliptic differential eguations, Comm. Pure Appl. Math. 14 (1961) 577-591.
36. J. Moser, A Harnack ineguality for parabolic differential eguations, Comm. Pure Appl. Math. 17 (1964) 101-134.
37. M. H. Protter &H. F. Weinberger, Maximum principles in differential equations, Prentice Hall, Englewood Cliffs, NJ, 1967.
38. J. P. Rosay &W. Rudin, Holomorphic maps from C” to C”, Trans. Amer. Math. Soc. 310 (1988) 47-86.
39. R. Schoen &S. T. Yau, Differential Geometry, Academia Sinica Press, Beijing, 1988.
40. W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223-301.
41. W. X. Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303-394.
42. W. X. Shi, Complete noncompact three-manifolds with nonnegative Ricci curvature, J. Differential Geom. 29 (1989) 353-360.
43. W. X. Shi, Ricci deformation of the metric on complete noncompact Kahler manifolds, Ph.D. thesis, Harvard University, 1990.
44. W. X. Shi, Complete noncompact Kahler manifolds with positive holomorphic bisectional curvature, Bull. Amer. Math. Soc. 23 (1990) 437-440.
45. J. Simons, On the transitivity of holnomy systems, Ann. of Math. 76 (1962) 213-234.
46. Y.T. Siu &S.T. Yau, Compact Kahler manifolds of positive bisectional curvature, Invent. Math. 59 (1980) 189-204.
47. G. Tian, On Calabi’s conjecture for complex surfaces with positive first Ghern class, Invent. Math. 101 (1990) 101-172.
48. S.T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation. I, Comm. Pure Appl. Math. 31 (1978) 339-411