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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)
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)
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:
added pointer to today’s
but haven’t really looked at what’s going on there
added pointer to today’s
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