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For the $2D$ Kahler manifold, the Ricci flow equation (which is also a one-loop RG equation for the $\sigma$-model on this space) can be written in the form $\frac{\partial^2 \Phi}{\partial u^2}=\frac{\partial \Phi}{\partial u} \frac{\partial \Phi}{\partial \tau}$, where $\Phi$ is related to the conformal factor of the metric, $\Omega(u, \tau)$. The solution gives the behaviour of $\Omega$ as we move along the RG time $\tau$, giving the scale-dependence of the $\sigma$-model QFT. The equation looks simple, so I suspect that it admits an explicit solution, but I can't find it.

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    $\begingroup$ Note that Yes or No questions tend to be a poor fit for this Q&A site. Can you also re-phrase this question so it isn't so much about "Do this work for me" and more-so about understanding the Ricci flow or something that is preventing you from solving the problem on your own? $\endgroup$
    – Kyle Kanos
    Nov 17, 2016 at 14:05
  • $\begingroup$ To be clear, you are explicitly and only asking for solutions about the PDE $\frac{\partial^2 \Phi}{\partial u^2}=\frac{\partial \Phi}{\partial u} \frac{\partial \Phi}{\partial \tau}$? $\endgroup$ Nov 17, 2016 at 15:44
  • $\begingroup$ @EmilioPisanty Yep. $\endgroup$ Nov 17, 2016 at 15:49

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Actually, it seems that this equation has an analytical solution only for a few initial metrics, which are presented, say, in this paper.

The analytical solution exists, if the boundary conditions allow the solution of the form $\omega (u, \tau)=a(\tau)+b(\tau) \psi(u)$ for $\psi(u)=\mathrm{exp} \left[ \pm \lambda u \right]$, $\mathrm{cosh} \left[\lambda u+A \right]$, $\mathrm{sinh} \left[\lambda u+A \right]$, $\mathrm{cos} \left[\lambda u+A \right]$, and $\frac{1}{\omega}=\frac{\partial \Phi}{\partial u}$.

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  • $\begingroup$ It would be good if you could summarise the results on the SE. $\endgroup$
    – JamalS
    Nov 26, 2016 at 21:47

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