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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded briefly something to [[Ricci flow]], amplifying that it is the [[gradient flow]] of the action functional of [[dilaton gravity]] (equivalently: the renormalization group flow of the string-sigma-model coupled to gravity and dilaton)

   added briefly something to Ricci flow, amplifying that it is the gradient flow of the action functional of dilaton gravity (equivalently: the renormalization group flow of the string-sigma-model coupled to gravity and dilaton)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 2nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded more references at _[[Ricci flow]]_ (a stub!). First of all I added Perelman's article where he demonstrates the monotonicity of the flow for gravity-dilaton background. Then I added a bunch of references which discuss the relation to (the identification with) the bosonic string sigma model: *** 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](http://kasperolsen.wordpress.com/2010/03/19/from-polyakov-to-perelman-history-of-the-ricci-flow-and-a-millennium-prize/)_ * E. Woolgar, _Some Applications of Ricci Flow in Physics_, Can.J.Phys.86:645,2008 ([arXiv:0708.2144](http://arxiv.org/abs/0708.2144)) * Mauro Carfora, _Renormalization Group and the Ricci flow_ ([arXiv:1001.3595](http://arxiv.org/abs/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](http://arxiv.org/abs/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](http://arxiv.org/abs/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](http://arxiv.org/abs/hep-th/0612296))

   added more references at Ricci flow (a stub!). First of all I added Perelman’s article where he demonstrates the monotonicity of the flow for gravity-dilaton background.

   Then I added a bunch of references which discuss the relation to (the identification with) the bosonic string sigma model:

   ---

   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)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 2nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexand I have slightly expanded the [Idea-section](http://ncatlab.org/nlab/show/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 gauge field|background fields]] containing [[gravity]] and [[dilaton]] (reviewed e.g. in [Woolgra 07](Woolgra07), [Carfora 10](#Carfora10), see also the introduction of [Tseytlin 06](#Tseytlin06)). In ([Perelman 02](#Perelman02)) 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](#RubinsteinSinclair) a typical such Ricci flow with pinchings is visualized as follows: <img src="http://i.stack.imgur.com/FmUuW.png" alt="" />

   and I have slightly expanded the Idea-section:

   ---

   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:

   

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 30th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * Alexander Frenkel, Petr Horava, Stephen Randall, _Topological Quantum Gravity of the Ricci Flow_ ([arXiv:2010.15369](https://arxiv.org/abs/2010.15369)) but haven't really looked at what's going on there <a href="https://ncatlab.org/nlab/revision/diff/Ricci+flow/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ricci+flow/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Ricci+flow">current</a>

   added pointer to today’s

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

   but haven’t really looked at what’s going on there

   diff, v6, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * Alexander Frenkel, [[Petr Horava]], Stephen Randall, _Perelman's Ricci Flow in Topological Quantum Gravity_ ([arXiv:2011.11914](https://arxiv.org/abs/2011.11914)) <a href="https://ncatlab.org/nlab/revision/diff/Ricci+flow/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ricci+flow/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Ricci+flow">current</a>

   added pointer to today’s

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

   diff, v7, current