Two dimensional Ricci flow is really easy to manage. In this case the equation takes a very simple form and a wealth of results can be extracted. As you know from my preceding posts, I have been able to prove in a rigorous way that in this case the Ricci flow arises from Brownian motion (see here). So, the equation for Einstein manifolds in this case takes the very simple form, being a constant, that is also the equation for a Ricci soliton. This equation is rather well-knwon to physicists as is the equation of 2d Einstein gravity. This equation is nothing else than Liouville equation
that admits an exact solution notwithstanding being non-linear. There is an unexpected application of all this machinery of Riemann geometry to the case of statistics. Statistics has a wide body of application fields as radar tracking, digital communications and so on. Then, any new result about can be translated into a wealthy number of applications.
The problem one meets in this case is that of parameter estimation of a given probability distribution. For a sample of measured data the question is to determine the best probability distribution with respect to the spread of the data themselves with a proper choice of the parameters. A known result in this area is the so called Cramer-Rao bound. This inequality gives limit for the optimality of the chosen estimators of the data entering into the distribution. The result I have found is that, for a probability distribution with two parameters, an infinite class of optimal estimators exists that are all efficient. These estimators are given by the solution of Liouville equation! The result can be extended to the n-dimensional case granted the existence of isothermal coordinates that are the conformal ones.
This result arises from the deep link between differential geometry and statistics that was put forward by Calayampudi Radhakrishna Rao. My personal interest in this matter was arisen working in radar tracking but one can think on a large number of other areas. I should say, as a final consideration, that the work of Hamilton and Perelman can have a deep impact in a large body of our knowledge. We are just at the beginning.
Today I have read recent changes to DispersiveWiki. This is a beautiful site about differential equations that is maintained at University of Toronto by Jim Colliander and has notable contributors as the Fields medallist Terence Tao. Terry introduced a new page about Liouville’s equation as he got involved with it in a way you can read here. Physicists working on quantum gravity has been aware of this equation since eighties as it is the equation of two-dimensional quantum gravity and comes out quite naturally in string theory. A beautiful paper about quantum field theory of Liouville equation is due to Roman Jackiw and one of his collaborators Eric D’Hoker (see here). But what people could have overlooked is that Liouville’s equation is the equation of the Ricci soliton in two dimensions. The reason is that in this case a set of isothermal coordinates can always be found and the metric is always conformal, that is
being the Euclidean metric. The Ricci tensor takes here a quite simple form
being . Then the Ricci flow is
and finally for the Ricci soliton one has
being a constant. After a simple rescaling we are left with the Euclidean Liouville’s equation
Turning back to the Jackiw and D’Hoker paper, we can see that a 2D gravity theory emerges naturally as the equilibrium (Ricci soliton) solution of a Fokker-Planck (Ricci flow) equation. This scenario seems a beautiful starting point to build an understanding of quantum gravity. I am still thinking about a lot and I will put all this on a paper one day.
These days I am looking at all this area of mathematical research born with Richard Hamilton and put at maturity with the works of Grisha Perelman. As all of you surely know the conclusion was that the Thurston conjecture, implying Poincare’ conjecture, is a theorem. These results present the shocking aspect of a deep truth waiting for an understanding by physicists and, I think that this comes out unexpectedly, statisticians (do you know Fischer information matrix and Cramer-Rao bound?).
One of the most shocking concept mathematicians introduced working with Ricci flow is a Ricci soliton. I will use some mathematics to explain this. A Ricci flow is given by
a Ricci soliton is a metric solving the equation
where I have used an awkward notation for the Lie derivative along a field X but if this field is a scalar than one has a gradient soliton. I think that all of you will recognize these equations that for a Lorentzian metric are just Einstein equations in vacuum with a cosmological constant! Now, I have found a beautiful paper about all this question on arxiv (see here). This paper gives the first meaningful application to physics of this striking concept. Ricci solitons are resembling a kind of behavior of the metric under the flow that can be expanding, collapsing or static depending on the cosmological constant.
As time goes by we learn something deeper about Einstein equations. Their very nature seems rooted in quite recent concepts coming from differential geometry and it is my personal view that whatever quantum gravity theory we will formulate, these are the questions we have to cope with.