Recent developments on the Ricci flow

Cao, Huai-Dong; Chow, Bennett

doi:10.1090/S0273-0979-99-00773-9

Recent developments on the Ricci flow
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Bull. Amer. Math. Soc. 36 (1999), 59-74 Request permission

Abstract:

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

References

Uwe Abresch and Wolfgang T. Meyer, Injectivity radius estimates and sphere theorems, Comparison geometry (Berkeley, CA, 1993–94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 1–47. MR 1452866, DOI 10.2977/prims/1195166274
Michael T. Anderson, Extrema of curvature functionals on the space of metrics on $3$-manifolds, Calc. Var. Partial Differential Equations 5 (1997), no. 3, 199–269. MR 1438146, DOI 10.1007/s005260050066
Michael T. Anderson, Scalar curvature and geometrization conjectures for $3$-manifolds, Comparison geometry (Berkeley, CA, 1993–94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 49–82. MR 1452867
Shigetoshi Bando, On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature, J. Differential Geom. 19 (1984), no. 2, 283–297. MR 755227
J. Bartz, M. Struwe, and R. Ye, A new approach to the Ricci flow on $S^2$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 3, 475–482. MR 1310637
Huai Dong Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), no. 2, 359–372. MR 799272, DOI 10.1007/BF01389058
Huai Dong Cao, On Harnack’s inequalities for the Kähler-Ricci flow, Invent. Math. 109 (1992), no. 2, 247–263. MR 1172691, DOI 10.1007/BF01232027
H.-D. Cao, Limits of solutions to the Kähler-Ricci flow, J. Differential Geom. 45 (1997) 257-272.
Huai-Dong Cao, Existence of gradient Kähler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994) A K Peters, Wellesley, MA, 1996, pp. 1–16. MR 1417944
H.-D. Cao and R. S. Hamilton, Gradient Kähler-Ricci solitons and periodic orbits, Comm. Anal. Geom. (to appear).
Haiwen Chen, Pointwise $\frac 14$-pinched $4$-manifolds, Ann. Global Anal. Geom. 9 (1991), no. 2, 161–176. MR 1136125, DOI 10.1007/BF00776854
Bennett Chow, The Ricci flow on the $2$-sphere, J. Differential Geom. 33 (1991), no. 2, 325–334. MR 1094458
Bennett Chow and Sun-Chin Chu, A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow, Math. Res. Lett. 2 (1995), no. 6, 701–718. MR 1362964, DOI 10.4310/MRL.1995.v2.n6.a4
Jeff Cheeger and Mikhael Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986), no. 3, 309–346. MR 852159
Dennis M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), no. 1, 157–162. MR 697987
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419, DOI 10.1090/conm/071/954419
Richard S. Hamilton, An isoperimetric estimate for the Ricci flow on the two-sphere, Modern methods in complex analysis (Princeton, NJ, 1992) Ann. of Math. Stud., vol. 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 191–200. MR 1369139, DOI 10.1080/09502389500490321
R. S. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. (1998) to appear.
R. S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1-92.
Richard S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993), no. 1, 225–243. MR 1198607
Richard S. Hamilton, Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1993), no. 1, 1–11. MR 1231700
Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47–62. MR 806701
James Isenberg and Martin Jackson, Ricci flow of locally homogeneous geometries on closed manifolds, J. Differential Geom. 35 (1992), no. 3, 723–741. MR 1163457
Thomas Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301–307. MR 1249376, DOI 10.1016/0926-2245(93)90008-O
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
Christophe Margerin, Un théorème optimal pour le pincement faible en dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 17, 877–880 (French, with English summary). MR 870911
Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227. MR 924677, DOI 10.2307/1971420
Ngaiming Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), no. 2, 179–214. MR 925119
Seiki Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 343–352. MR 840284, DOI 10.1090/pspum/044/840284
Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. MR 1001277
Wan-Xiong Shi, Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 437–440. MR 1044171, DOI 10.1090/S0273-0979-1990-15954-3
Wan-Xiong Shi, Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differential Geom. 45 (1997), no. 1, 94–220. MR 1443333
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0