IMPORTANT NOTICE

The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at cust-serv@ams.org or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).

 

Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Recent developments on the Ricci flow


Authors: Huai-Dong Cao and Bennett Chow
Journal: Bull. Amer. Math. Soc. 36 (1999), 59-74
MSC (1991): Primary 58G11; Secondary 53C21, 35K55
DOI: https://doi.org/10.1090/S0273-0979-99-00773-9
MathSciNet review: 1655479
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article reports recent developments of the research on Hamilton's Ricci flow and its applications.


References [Enhancements On Off] (What's this?)

  • [AM] 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
  • [A1] M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds, Calc. Var. 5 (1997) 199-269. MR 98a:58041
  • [A2] 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
  • [Ba] S. Bando, On three-dimensional compact Kähler manifolds of nonnegative bisectional curvature, J.D.G. 19 (1984) 283-297. MR 86i:53042
  • [BSY] J. Bartz, M. Struwe, and R. Ye, A new approach to the Ricci flow on $S^{2}$, Annali de Scuola Normale Superiore di Pisa 21 (1994) 475-482. MR 95i:53039
  • [Ca1] H.-D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985) 359-372. MR 87d:58051
  • [Ca2] H.-D. Cao, On Harnack's inequalities for the Kähler-Ricci flow, Invent. Math. 109 (1992) 247-263. MR 93f:58227
  • [Ca3] H.-D. Cao, Limits of solutions to the Kähler-Ricci flow, J. Differential Geom. 45 (1997) 257-272. CMP 97:13
  • [Ca4] H.-D. Cao, Existence of gradient Kähler-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
  • [CH] H.-D. Cao and R. S. Hamilton, Gradient Kähler-Ricci solitons and periodic orbits, Comm. Anal. Geom. (to appear).
  • [C] H. Chen, Pointwise quarter-pinched $\mathit{4}$-manifolds, Ann. Global Anal. Geom. 9 (1991) 161-176. MR 93b:53028
  • [Ch] B. Chow, The Ricci flow on the $\mathit{2}$-sphere, J. Differential Geom. 33 (1991) 325-334. MR 92d:53036
  • [CC] 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
  • [CG] 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
  • [De] 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
  • [ES] J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160. MR 29:1603
  • [Ha1] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306. MR 84a:53050
  • [Ha2] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153-179. MR 87m:53055
  • [Ha3] R. S. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics 71 (1988), 237-261. MR 89i:53029
  • [Ha4] 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
  • [Ha5] R. S. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. (1998) to appear.
  • [Ha6] R. S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1-92. CMP 97:14
  • [Ha7] R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993) 225-243. MR 93k:58052
  • [Ha8] R. S. Hamilton, Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1993) 1-11. MR 94g:58043
  • [Ha9] R. S. Hamilton, Formation of singularities in the Ricci flow, Surveys in Diff. Geom. 2 (1995) 7-136, International Press, Boston. MR 97e:53075
  • [Hu] G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21(1985) 47-62. MR 86k:53059
  • [IJ] J. Isenberg and M. Jackson, Ricci flow of locally homogeneous geometries on closed manifolds, J. Diff. Geom. 35 (1992) 723-741. MR 93c:58049
  • [Iv] T. Ivey, Ricci solitons on compact three-manifolds, Diff. Geom. and its Appl. 3 (1993) 301-307. MR 94j:53048
  • [LY] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math. 156 (1986) 153-201. MR 87f:58156
  • [Ma] 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
  • [MM] 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
  • [Mo] N. Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988) 179-214. MR 89d:53115
  • [Ni] 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
  • [Sc] R. Schoen, Conformal deformation of a Riemannian to constant scalar curvature, J. Differential Geom. 20 (1984) 479-495. MR 86i:58137
  • [SY] R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topology of $\mathit{3}$-manifolds with nonnegative scalar curvature, Ann. Math. (1979) 110 127-142. MR 81k:58029
  • [S] P. Scott, The geometries of $\mathit{3}$-manifolds, Bull. London Math. Soc. 15 (1983) 401-487. MR 84m:57009
  • [Sh1] 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
  • [Sh2] W. X. Shi, Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature, Bull. Amer. Math. Soc. 23 (1990) 437-440. MR 91e:53069
  • [Sh3] W. X. Shi, Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differential Geom. 45 (1997) 94-220. MR 98d:53099
  • [T] W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357-381. MR 83h:57019

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 58G11, 53C21, 35K55

Retrieve articles in all journals with MSC (1991): 58G11, 53C21, 35K55


Additional Information

Huai-Dong Cao
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843
Email: cao@math.tamu.edu

Bennett Chow
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, MN 55455
Email: bchow@math.umn.edu

DOI: https://doi.org/10.1090/S0273-0979-99-00773-9
Received by editor(s): June 17, 1997
Received by editor(s) in revised form: October 15, 1998
Additional Notes: Authors partially supported by the NSF
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society