Mathematics > Differential Geometry
The entropy formula for the Ricci flow and its geometric applications
Grisha Perelman
(Submitted on 11 Nov 2002)
We present a monotonic expression for the Ricci flow, valid in all dimensions and without 
curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. 
Several geometric applications are given. In particular, (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. We also 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, making use of earlier results on collapsing
 with local lower curvature bound.