Calabi flow

Remarks

Example: if $\omega_0$ has constant scalar curvature, such as a Kähler–Einstein metric or a flat metric on a complex torus, then $S(0) = \underline{S}$, and the constant family $\varphi_t = 0$ solves the flow. Non-example: a family $\omega_t = (1 + t)\omega_0$ leaves the Kähler class of $\omega_0$, so it is not of the form $\omega_0 + i\partial\bar\partial\varphi_t$ and is not a Calabi flow. In local coordinates the flow reads $\partial g_{j\bar k}/\partial t = \partial_j \partial_{\bar k} S$ for the metric coefficients $g_{j\bar k}$. Along the flow the Calabi energy $\int_M (S - \underline{S})^2\, \omega_\varphi^n$ does not increase.