The entropy formula for the Ricci flow and its geometric applications
Grisha Perelman
TL;DR
This work introduces a monotone entropy functional for the Ricci flow valid in all dimensions, recasting the flow as a gradient flow and yielding strong rigidity results such as no nontrivial breathers. It develops a robust framework—including an extended geometric formalism, reduced volume, L-length, and pseudo-locality—that yields local injectivity-radius control, noncollapsing theorems, and differential Harnack inequalities. These tools are then applied to ancient solutions and 3-manifolds, showing blow-up limits are gradient shrinking solitons and enabling a path toward Thurston geometrization via a thick-thin decomposition and surgical procedures. Collectively, the results provide deep structural insight into Ricci flow dynamics and lay groundwork for a rigorous geometric classification of 3-manifolds through Ricci flow techniques.
Abstract
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.