How to derive the cigar soliton solution to the Ricci flow equation?

Question

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form

$${\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}}$$

I am starting from a metric with he form

$${\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{f \left( t,x,y \right) }}$$

and from the Ricci flow equations $dg_{ij}/dt = -2 R_{ij}$, I am obtaining

$$-{\frac {\partial }{\partial t}}f \left( t,x,y \right) = \left( { \frac {\partial }{\partial y}}f \left( t,x,y \right) \right) ^{2}- \left( {\frac {\partial ^{2}}{\partial {y}^{2}}}f \left( t,x,y \right) \right) f \left( t,x,y \right) + \left( {\frac {\partial }{ \partial x}}f \left( t,x,y \right) \right) ^{2}- \left( {\frac { \partial ^{2}}{\partial {x}^{2}}}f \left( t,x,y \right) \right) f \left( t,x,y \right)$$

I am looking for a solution of the form $f(t,x,y)=F(t)+g(x)+h(y)$. Then I obtain

$$f \left( t,x,y \right) ={\frac {{C_{{2}}}^{2}}{C_{{1}}}}+{{\rm e}^{2\, C_{{1}}t}}C_{{3}}+{\frac {1}{2}}C_{{1}}{x}^{2}+C_{{2}}x+{\frac {1}{2}}C_{{1}}{y}^{2}+C_{ {2}}y$$

Please let me know what other conditions it is necessary to apply with the aim to obtain the cigar soliton solution. Many thanks.

This post imported from StackExchange Physics at 2014-10-09 19:43 (UTC), posted by SE-user Juan Ospina

Comments

I am by no means familiar with these things. However, if you required that the solution is $O(2)$ invariant this would rule out $C_2$. As a weaker requirement to get the same result you may suppose that the metric is invarian under reflection $x \to -x$ (or $y\to -y$). I cannot see any evident requirement to fix $C_2$ and $C_3$.