Ricci solitons

22/01/2009

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

$\frac{\partial g_{ik}}{\partial t}= -2R_{ik}$

a Ricci soliton is a metric solving the equation

$R_{ik}-L_X g_{ik} = \Lambda g_{ik}$

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.