Type II collapsing of maximal solutions to the Ricci flow in ${\mathbb{R}}^2$

Abstract

We consider the initial value problem $u_t=\mathrm{\Delta }\log u$, $u(x,0)=u_0(x)⩾0$ in ${\mathbb{R}}^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u(\mathrm{d}{x}_1^2+\mathrm{d}{x}_2^2)$ by Ricci curvature. It is well known that the maximal solution $u$ vanishes identically after time $T=\frac{1}{4\pi }{\int }_{{\mathbb{R}}^2}{u}_0$. Assuming that $u_0$ is radially symmetric and satisfies some additional constraints, we describe precisely the Type II collapsing of $u$ at time $T$: we show the existence of an inner region with exponentially fast collapsing and profile, up to proper scaling, a 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 [J.R. King, Self-similar behavior for the equation of fast nonlinear diffusion, Philos. Trans. R. Soc. London Ser. A 343 (1993) 337–375].