The entropy formula for the Ricci flow and its geometric applications

TL;DR

This paper introduces a new entropy formula for Ricci flow applicable in all dimensions, leading to significant geometric insights and constraints on the evolution of Riemannian metrics, with implications for Thurston's geometrization conjecture.

Contribution

A novel monotonic entropy formula for Ricci flow that applies universally and yields new geometric applications and constraints.

Findings

Ricci flow has no nontrivial periodic orbits aside from fixed points

Injectivity radius is controlled during finite-time singularities

Ricci flow cannot rapidly increase curvature in almost Euclidean regions

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.