The entropy formula for the Ricci flow and its geometric applications

By G. Perelman. 2002

3,287 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/6962af2ce217d906d7e907d7dab7369820befc17

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Uniqueness and pseudolocality theorems of the mean curvature flow (2007, methodology): “man’s point-picking technique [ 13 ] to choose another point which controls nearby points in its scale.”
Eigenvalues of on manifolds with nonnegative curvature operator (2007, result): “As an application, Perelman was able to show that there are no nontrivial steady or expanding breathers on compact manifolds.”
Ricci solitons and concurrent vector fields (2014, methodology): “It was proved in [19] that if (M, g, ξ, λ) is a compact Ricci soliton, the potential field ∗ Balkan Journal of Geometry and Its Applications, Vol.”
Recent progress on the Poincaré conjecture and the classification of 3-manifolds (2004, background): “More recently, building on this work, Perelman, in a series of three preprints [15, 16, 17], has claimed to surmount all of the various technical, geometric and analytic difficulties of Hamilton’s program.”
On stability and the convergence of the Kähler-Ricci flow (2006, methodology): “and K¨ahler forms, is actually the version of the Gromov compactness theorem established by Hamilton [20], where C k uniform bounds on the curvature are assumed, instead of just C 0 bounds as in the original version of Gromov, Pet er s [25], Greene and Wu [16]”