nLab
Ricci flow

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Riemannian geometry

Contents

Idea

Ricci flow is the gradient flow of the the action functional of dilaton gravity: the Einstein-Hilbert action coupled to a dilaton field.

(Equivalently: it is the renormalization group flow of the string sigma-model for background fields containing gravity and dilaton.)

From Rubinstein-Sinclair:

References

The identification of Ricci flow as a gradient flow of gravity+dilaton is due to

  • Perelman,

and was a key step in his completion of Hamilton’s program for how to prove the Poincare conjecture.

A quick survey is in the slides

  • Annibale Magni, Perelman’s dilaton (pdf).

A detailed survey is in

  • Terry Tao, Ricci flow as a gradient flow, log-Sobolev inequalities, and Perelman entropy (blog post)

  • J. Hyam Rubinstein and Robert Sinclair. “Visualizing Ricci Flow of Manifolds of Revolution”, Experimental Mathematics v. 14 n. 3, pp. 257–384.

Revised on November 19, 2013 11:59:32 by Urs Schreiber (77.251.114.72)