# nLab Ricci flow

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

## Applications

#### Riemannian geometry

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)