# Calabi flow

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In the mathematical fields of [differential geometry](/source/differential_geometry) and [geometric analysis](/source/geometric_analysis), the '''Calabi flow''' is a [geometric flow](/source/geometric_flow) which deforms a [Kähler metric](/source/K%C3%A4hler_metric) on a [complex manifold](/source/complex_manifold). Precisely, given a [Kähler manifold](/source/K%C3%A4hler_manifold) {{mvar|M}}, the Calabi flow is given by:
:<math>\frac{\partial g_{\alpha\overline{\beta}}}{\partial t}=\frac{\partial^2 R^g}{\partial z^\alpha\partial\overline{z}^\beta}</math>,
where {{mvar|g}} is a mapping from an open interval into the collection of all Kähler metrics on {{mvar|M}}, {{math|''R''<sup>''g''</sup>}} is the [scalar curvature](/source/scalar_curvature) of the individual Kähler metrics, and the indices {{math|α, β}} correspond to arbitrary holomorphic coordinates {{math|''z''<sup>α</sup>}}. This is a fourth-order geometric flow, as the right-hand side of the equation involves fourth [derivative](/source/derivative)s of {{mvar|g}}.

The Calabi flow was introduced by [Eugenio Calabi](/source/Eugenio_Calabi) in 1982 as a suggestion for the construction of [extremal Kähler metric](/source/extremal_K%C3%A4hler_metric)s, which were also introduced in the same paper. It is the gradient flow of the ''{{visible anchor|Calabi functional}}''; extremal Kähler metrics are the [critical points](/source/Critical_point_(mathematics)) of the Calabi functional.

A convergence theorem for the Calabi flow was found by Piotr Chruściel in the case that {{mvar|M}} has complex dimension equal to one. [Xiuxiong Chen](/source/Xiuxiong_Chen) and others have made a number of further studies of the flow, although as of 2020 the flow is still not well understood.

==References==
* Eugenio Calabi. Extremal Kähler metrics. [Ann. of Math. Stud.](/source/Ann._of_Math._Stud.) 102 (1982), pp. 259–290. Seminar on Differential Geometry. [Princeton University Press](/source/Princeton_University_Press) (PUP), [Princeton, N.J.](/source/Princeton%2C_N.J.)
* E. Calabi and X.X. Chen. The space of Kähler metrics. II. [J. Differential Geom.](/source/J._Differential_Geom.) 61 (2002), no. 2, 173–193.
* X.X. Chen and W.Y. He. On the Calabi flow. Amer. J. Math. 130 (2008), no. 2, 539–570.
* Piotr T. Chruściel. Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation. Comm. Math. Phys. 137 (1991), no. 2, 289–313.

{{DEFAULTSORT:Calabi Flow}}
Category:Geometric flow
Category:Partial differential equations
Category:String theory

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Adapted from the Wikipedia article [Calabi flow](https://en.wikipedia.org/wiki/Calabi_flow) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Calabi_flow?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.