Type II extinction profile of maximal solutions to the Ricci flow in $\R^2$
Abstract
We consider the initial value problem $u_t = \Delta \log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4\pi} \int_{\R^2} u_0 $. Assuming that $u_0$ is compactly supported we describe precisely the Type II vanishing behavior of $u$ at time $T$: we show the existence of an inner region with exponentially fast vanishing profile, which is, up to proper scaling, a {\em soliton cigar solution}, and the existence of an outer region of persistence of a logarithmic cusp. This is the only Type II singularity which has been shown to exist, so far, in the Ricci Flow in any dimension. It recovers rigorously formal asymptotics derived by J.R. King \cite{K}.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606288
 Bibcode:
 2006math......6288D
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 35K55;
 35B40