nLab Ricci flow

Idea

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

Equivalently it is the renormalization group flow of the bosonic string sigma-model for background fields containing gravity and dilaton (reviewed e.g. in Woolgra 07, Carfora 10, see also the introduction of Tseytlin 06).

In (Perelman 02) Ricci flow for dilaton gravity in 3d was shown to enjoy sufficient monotonicity properties such as to complete Richard Hamilton‘s proof of the Poincaré conjecture. Hamilton’s strategy was to equip a compact simply connected 3d manifold with a Riemannian metric, argue that its Ricci flow will, after many “pinchings” (points where the metric degenerates) produce a collection of 3-spheres and conclude that therefor the original manifold must have been a 3-sphere, too. The technical problem is to control the number of pinchings, which may occur rapidly. Adding also the dilaton field turns out not to change the qualitative nature of the flow but make it “monotone enough” to control the pinching.

In Rubinstein-Sinclair a typical such Ricci flow with pinchings is visualized as follows:

References

General

A quick survey is in the slides

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

and a detailed survey is in

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

Visualization is in

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

The monotonicity of the Ricci flow for the string sigma-model in dilaton gravity background was established in

• Grigori Perelman, The entropy formula for the Ricci flow and its geometric applications (arXiv:math/0211159)

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

As sigma-model renormalization group flow

The identification of Ricci flow with the renormalization group flow of the bosonic string sigma-model is reviewed for instance in

• Kasper Olsen, From Polyakov to Perelman

• E. Woolgar, Some Applications of Ricci Flow in Physics, Can.J.Phys.86:645,2008 (arXiv:0708.2144)

• Mauro Carfora, Renormalization Group and the Ricci flow (arXiv:1001.3595)

and discussed in more detail for instance in

• T Oliynyk, V Suneeta, E Woolgar, Irreversibility of World-sheet Renormalization Group Flow, Phys.Lett. B610 (2005) 115-121 (arXiv:hep-th/0410001)

• T Oliynyk, V Suneeta, E Woolgar, A Gradient Flow for Worldsheet Nonlinear Sigma Models, Nucl.Phys. B739 (2006) 441-458 (arXiv:hep-th/0510239)

• Arkady Tseytlin, On sigma model RG flow, “central charge” action and Perelman’s entropy, Phys.Rev.D75:064024,2007 (arXiv:hep-th/0612296)

See also at string theory – References – Fields medal work related to string theory.

See also

• Alexander Frenkel, Petr Horava, Stephen Randall, Topological Quantum Gravity of the Ricci Flow (arXiv:2010.15369)

• Alexander Frenkel, Petr Horava, Stephen Randall, Perelman’s Ricci Flow in Topological Quantum Gravity (arXiv:2011.11914)