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39 pages, February 1, 2008

Abstract:

"The entropy formula for the Ricci flow and its geometric applications." Author's statements: "Several geometric applications are given.(1) Ricci flow, considered on the space of Riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away." The author wrote also that he was able to: "verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three--manifolds, and give a sketch of an eclectic proof of this conjecture" The Ricci flow has previously been discussed in quantum field theory, as an approximation to the renormalization group (RG) flow for the two--dimensional nonlinear sigma--model(by Gaw; and references therein). Thus, Perelman also proposed in 2002 (v.1) that the Ricci flow be defined as a gradient flow; than he showed that the Ricci flow-- considered as a dynamical system on the space of Riemannian metrics modulo diffeomorphisms and scaling-- has no nontrivial periodic orbits.

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