Recent developments on the Ricci flow

Bulletin of the American Mathematical Society

Volumn 36, Issue 1, 1999, Pages 59-74

  Cao, Huai Dong ^a   Chow, Bennett ^b  

^a TEXAS A AND M UNIVERSITY   (United States)

^b University of Minnesota   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033477993     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0273-0979-99-00773-9     Document Type: Article

References (85)

1. U. Abresch and W.T. Meyer, Injectivity radius estimates and sphere theorems, in Comparison Geometry, MSRI Publications, vol. 30 (1997) 1-47, Cambridge Univ. Press. MR 98e:53052
2. U. Abresch and W.T. Meyer, Injectivity radius estimates and sphere theorems, in Comparison Geometry, MSRI Publications, vol. 30 (1997) 1-47, Cambridge Univ. Press. MR 98e:53052
3. M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds, Calc. Var. 5 (1997) 199-269. MR 98a:58041
4. M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds, Calc. Var. 5 (1997) 199-269. MR 98a:58041
5. M. Anderson, Scalar curvature and geometrization conjectures for 3-manifolds, in Comparison Geometry, MSRI Publications, vol. 30 (1997) 49-82, Cambridge Univ. Press. MR 98e:58050
6. M. Anderson, Scalar curvature and geometrization conjectures for 3-manifolds, in Comparison Geometry, MSRI Publications, vol. 30 (1997) 49-82, Cambridge Univ. Press. MR 98e:58050
7. S. Bando, On three-dimensional compact Kähler manifolds of nonnegative bisectional curvature, J.D.G. 19 (1984) 283-297. MR 86i:53042
8. S. Bando, On three-dimensional compact Kähler manifolds of nonnegative bisectional curvature, J.D.G. 19 (1984) 283-297. MR 86i:53042
9. 2, Annali de Scuola Normale Superiore di Pisa 21 (1994) 475-482. MR 95i:53039
10. 2, Annali de Scuola Normale Superiore di Pisa 21 (1994) 475-482. MR 95i:53039
11. H.-D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985) 359-372. MR 87d:58051
12. H.-D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985) 359-372. MR 87d:58051
13. H.-D. Cao, On Harnack's inequalities for the Kähler-Ricci flow, Invent. Math. 109 (1992) 247-263. MR 93f:58227
14. H.-D. Cao, On Harnack's inequalities for the Kähler-Ricci flow, Invent. Math. 109 (1992) 247-263. MR 93f:58227
15. H.-D. Cao, Limits of solutions to the Kähler-Ricci flow, J. Differential Geom. 45 (1997) 257-272. CMP 97:13
16. H.-D. Cao, Existence of gradient Köhler-Ricci solitons, Elliptic and parabolic methods in geometry, B. Chow, R. Gulliver, S. Levy, J. Sullivan ed., AK Peters (1996) 1-16. MR 98a:53058
17. H.-D. Cao, Existence of gradient Köhler-Ricci solitons, Elliptic and parabolic methods in geometry, B. Chow, R. Gulliver, S. Levy, J. Sullivan ed., AK Peters (1996) 1-16. MR 98a:53058
18. H.-D. Cao and R. S. Hamilton, Gradient Kähler-Ricci solitons and periodic orbits, Comm. Anal. Geom. (to appear).
19. H. Chen, Pointwise quarter-pinched 4-manifolds, Ann. Global Anal. Geom. 9 (1991) 161-176. MR 93b:53028
20. H. Chen, Pointwise quarter-pinched 4-manifolds, Ann. Global Anal. Geom. 9 (1991) 161-176. MR 93b:53028
21. B. Chow, The Ricci flow on the 2-sphere, J. Differential Geom. 33 (1991) 325-334. MR 92d:53036
22. B. Chow, The Ricci flow on the 2-sphere, J. Differential Geom. 33 (1991) 325-334. MR 92d:53036
23. B. Chow and S.-C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow, Math. Research Letters 2 (1995) 701-718. MR 97f:53063
24. B. Chow and S.-C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow, Math. Research Letters 2 (1995) 701-718. MR 97f:53063
25. J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, II., J. Differential Geom. 23 (1986) 309-346, 32 (1990) 269-298. MR 87k:53087; MR 92a:53066
26. J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, II., J. Differential Geom. 23 (1986) 309-346, 32 (1990) 269-298. MR 87k:53087; MR 92a:53066
27. J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, II., J. Differential Geom. 23 (1986) 309-346, 32 (1990) 269-298. MR 87k:53087; MR 92a:53066
28. J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, II., J. Differential Geom. 23 (1986) 309-346, 32 (1990) 269-298. MR 87k:53087; MR 92a:53066
29. D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983) 157-162; ibid., improved version, to appear in Selected Papers on the Ricci Flow, ed. H.-D. Cao, B. Chow, S.-C. Chu, and S.-T. Yau, International Press. MR 85j:53050
30. D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983) 157-162; ibid., improved version, to appear in Selected Papers on the Ricci Flow, ed. H.-D. Cao, B. Chow, S.-C. Chu, and S.-T. Yau, International Press. MR 85j:53050
31. D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983) 157-162; ibid., improved version, to appear in Selected Papers on the Ricci Flow, ed. H.-D. Cao, B. Chow, S.-C. Chu, and S.-T. Yau, International Press. MR 85j:53050
32. J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160. MR 29:1603
33. J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160. MR 29:1603
34. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306. MR 84a:53050
35. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306. MR 84a:53050
36. R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153-179. MR 87m:53055
37. R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153-179. MR 87m:53055
38. R. S. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics 71 (1988), 237-261. MR 89i:53029
39. R. S. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics 71 (1988), 237-261. MR 89i:53029
40. R. S. Hamilton, An isoperimetric estimate for the Ricci flow on surfaces, in Modern Methods in Complex Analysis, The Princeton conference in honor of Gunning and Kohn, pp. 191-200, ed. T. Bloom, etal., Annals of Math. Studies 137, Princeton Univ. Press (1995). MR 96k:53059
41. R. S. Hamilton, An isoperimetric estimate for the Ricci flow on surfaces, in Modern Methods in Complex Analysis, The Princeton conference in honor of Gunning and Kohn, pp. 191-200, ed. T. Bloom, etal., Annals of Math. Studies 137, Princeton Univ. Press (1995). MR 96k:53059
42. R. S. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. (1998) to appear.
43. R. S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1-92. CMP 97:14
44. R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993) 225-243. MR 93k:58052
45. R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993) 225-243. MR 93k:58052
46. R. S. Hamilton, Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1993) 1-11. MR 94g:58043
47. R. S. Hamilton, Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1993) 1-11. MR 94g:58043
48. R. S. Hamilton, Formation of singularities in the Ricci flow, Surveys in Diff. Geom. 2 (1995) 7-136, International Press, Boston. MR 97e:53075
49. R. S. Hamilton, Formation of singularities in the Ricci flow, Surveys in Diff. Geom. 2 (1995) 7-136, International Press, Boston. MR 97e:53075
50. G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21(1985) 47-62. MR 86k:53059
51. G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21(1985) 47-62. MR 86k:53059
52. J. Isenberg and M. Jackson, Ricci flow of locally homogeneous geometries on closed manifolds, J. Diff. Geom. 35 (1992) 723-741. MR 93c:58049
53. J. Isenberg and M. Jackson, Ricci flow of locally homogeneous geometries on closed manifolds, J. Diff. Geom. 35 (1992) 723-741. MR 93c:58049
54. T. Ivey, Ricci solitons on compact three-manifolds, Diff. Geom. and its Appl. 3 (1993) 301-307. MR 94j:53048
55. T. Ivey, Ricci solitons on compact three-manifolds, Diff. Geom. and its Appl. 3 (1993) 301-307. MR 94j:53048
56. P. Li and S.-T. Yau. On the parabolic kernel of the Schrödinger operator, Acta. Math. 156 (1986) 153-201. MR 87f:58156
57. P. Li and S.-T. Yau. On the parabolic kernel of the Schrödinger operator, Acta. Math. 156 (1986) 153-201. MR 87f:58156
58. C. Margerin, A sharp theorem for weakly pinched 4-manifolds, C.R. Acad. Sci. Paris Serie 1 17 (1986) 303; Pointwise pinched manifolds are space forms, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986). MR 87k:53092; MR 87g:53063
59. C. Margerin, A sharp theorem for weakly pinched 4-manifolds, C.R. Acad. Sci. Paris Serie 1 17 (1986) 303; Pointwise pinched manifolds are space forms, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986). MR 87k:53092; MR 87g:53063
60. C. Margerin, A sharp theorem for weakly pinched 4-manifolds, C.R. Acad. Sci. Paris Serie 1 17 (1986) 303; Pointwise pinched manifolds are space forms, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986). MR 87k:53092; MR 87g:53063
61. C. Margerin, A sharp theorem for weakly pinched 4-manifolds, C.R. Acad. Sci. Paris Serie 1 17 (1986) 303; Pointwise pinched manifolds are space forms, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986). MR 87k:53092; MR 87g:53063
62. M. Micallef and J. D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988) 199-227. MR 89e:53088
63. M. Micallef and J. D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988) 199-227. MR 89e:53088
64. N. Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988) 179-214. MR 89d:53115
65. N. Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988) 179-214. MR 89d:53115
66. S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986) 343-352; On deformation of Riemannian metrics and manifolds with positive curvature operator, Lecture Notes in Math. 1201 (1986) 201-211. MR 87i:58032; MR 87k:53094
67. S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986) 343-352; On deformation of Riemannian metrics and manifolds with positive curvature operator, Lecture Notes in Math. 1201 (1986) 201-211. MR 87i:58032; MR 87k:53094
68. S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986) 343-352; On deformation of Riemannian metrics and manifolds with positive curvature operator, Lecture Notes in Math. 1201 (1986) 201-211. MR 87i:58032; MR 87k:53094
69. S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric Measure Theory Conference at Arcata, Proc. Symp. Pure Math. 44 (1986) 343-352; On deformation of Riemannian metrics and manifolds with positive curvature operator, Lecture Notes in Math. 1201 (1986) 201-211. MR 87i:58032; MR 87k:53094
70. R. Schoen, Conformal deformation of a Riemannian to constant scalar curvature, J. Differential Geom. 20 (1984) 479-495. MR 86i:58137
71. R. Schoen, Conformal deformation of a Riemannian to constant scalar curvature, J. Differential Geom. 20 (1984) 479-495. MR 86i:58137
72. R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topology of 3-manifolds with nonnegative scalar curvature, Ann. Math. (1979) 110 127-142. MR 81k:58029
73. R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topology of 3-manifolds with nonnegative scalar curvature, Ann. Math. (1979) 110 127-142. MR 81k:58029
74. P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983) 401-487. MR 84m:57009
75. P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983) 401-487. MR 84m:57009
76. W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223-301; Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303-394. MR 90i:58202; MR 90f:53080
77. W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223-301; Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303-394. MR 90i:58202; MR 90f:53080
78. W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223-301; Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303-394. MR 90i:58202; MR 90f:53080
79. W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223-301; Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303-394. MR 90i:58202; MR 90f:53080
80. W. X. Shi, Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature, Bull. Amer. Math. Soc. 23 (1990) 437-440. MR 91e:53069
81. W. X. Shi, Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature, Bull. Amer. Math. Soc. 23 (1990) 437-440. MR 91e:53069
82. W. X. Shi, Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differential Geom. 45 (1997) 94-220. MR 98d:53099
83. W. X. Shi, Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differential Geom. 45 (1997) 94-220. MR 98d:53099
84. W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357-381. MR 83h:57019
85. W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357-381. MR 83h:57019