Perelman' s monotonicity formula and applications

This article relies on [15] that the author wrote with Gang Tian and Xiaodong Wang. In view of Hamilton's important work on the Ricci flow and Perelman's paper on the Ricci flow where he developes the techniques that he will later use in completing Hamilton's program for the geometrization conjecture, there may be more interest in the area. We will also discuss the author's theorem which says that the curvature tensor stays uniformly bounded under the unnormalized Ricci flow in a finite time, if the curvatures are uniformly bounded. We will prove that in the case of a Kâhler-Ricci flow with uniformly bounded Ricci curvatures, for each sequence of flows g (ti+ t) for $t_irightarrowinfty$ there exists a subsequence of metrics converging to a solution to the flow outside a set of codimension 4.

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[1] M. Anderson: Convergence and rigidity of manifolds under Ricci curvature bounds, Invent.math. 102 (1990), 429-445 [2] H.D. Cao: Deformation of Kâhler metrics on compact Kâhler manifolds, Invent, math. 81,(1985), 359-372. [3] B. Chow: The Ricci flow on the 2-sphere, Journal of Differential Geometry, vol. 33 (1991),325-334.
[4] J. Cheeger, T.H. Colding: On the structure spaces with Ricci curvature bounded below I, J.Diff. Geom. 45 (1997), 406-480
[5] J. Cheeger, T.H. Colding, G.Tian: On the singularities of spaces with bounded Ricci curvature, Geom.funcft.anal. Vol. 12 (2002), 873-914
[6] D. DeTurck: Deforming metrics in the direction of their Ricci tensors, J.Differential Geom.18 (1983), no.l, 157-162.
[7] D. Glickenstein: Precompactness of solutions to the Ricci flow in the absence of injectivityradius estimates, preprint arXiv:math.DG/0211191 v2
[8] R. Hamilton: Three-manifolds with positive Ricci curvature, Journal of Differential Geometry 17 (1982) 225-306.
[9] R. Hamilton:. Four-manifolds with positive curvature operator, Journal of Differential Geometry 24 (1986) 255-306.
[10] R. Hamilton: The Ricci flow on surfaces, Contemporary Mathematics 71 (1988) 237-261.
[11] R. Hamilton: The formation of singularities in the Ricci flow; Surveys in Differential Geometry, vol. 2, International Press, Cambridge, MA (1995) 7-136.
[12] B. Kleiner, J. Lott: Notes on Perelman's paper.
[13] G. Perelman: The entropy formula for the Ricci flow and its geometric applications, preprint
[14] G. Perelman: unpublished work about the Kâhler-Ricci flow.
[15] N. Sesum, G. Tian, X. Wang: Notes on Perelman's paper on the entropy formula for theRicci flow and its geometric applications, in preparation.
[16] N. Sesum: Curvature tensor under the Ricci flow, preprint, arXiv: math.DG/0311397. [17] G. Tian, X. Zhu: A new holomorphic invariant and uniqueness of Kdhler-Ricci solitons, Comment. Math. Helv. 77 (2002), 1-29