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