---
tag: 13D1AB6RT8
kind: definition
name: Calabi flow
msc: ["53E40", "53C55"]
defines: 9FBJED00CP
uses: [CK1DCPC3D3, ZHX2VYQDZ3]
sources:
  - url: https://en.wikipedia.org/wiki/Calabi_flow
---

## Definition

Let $M$ be a compact complex manifold of complex dimension $n$ with a Kähler form $\omega_0$.
This is the form of a [[Hermitian metric|CK1DCPC3D3]] with $d\omega_0 = 0$.
For a smooth real function $\varphi$ on $M$ with $\omega_\varphi = \omega_0 + i \partial \bar\partial \varphi > 0$, let $S(\varphi)$ be the [[scalar curvature|ZHX2VYQDZ3]] of the Kähler metric $\omega_\varphi$.
Let $\underline{S}$ be the average of $S(\varphi)$ with respect to the volume form $\omega_\varphi^n$; it depends only on the Kähler class of $\omega_0$.
A smooth family $(\varphi_t)_{0 \le t < T}$ of such functions is a solution of the *Calabi flow* if
$$\frac{\partial \varphi_t}{\partial t} = S(\varphi_t) - \underline{S}.$$

## 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.