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.
