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Given a metric $g_{a\overline b}$ defined on a Kaheler manifold $K$, the Calabi flow is defined by the equation: $$\partial_u g_{a\overline b}=\frac{\partial^2 R}{\partial Z^a \partial Z^\overline b}$$

I know there is a set of conditions on the manifold $K$ at which the previous equation becomes a $Robinson \ Trautman$ equation. What are these conditions? Thanks for answers or suggestions.

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This depends on what you mean by a Robinson-Trautman equation. I learned that Calabi flow coincides with the Robinson-Trautman equations when the number of dimensions is 2.

A classic reference is Chrusciel, which you should consult for further information.

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