In the mathematical fields of differential geometry and geometric analysis, the Calabi flow is a geometric flow which deforms a Kähler metric on a complex manifold.
Precisely, given a Kähler manifold M, the Calabi flow is given by: where g is a mapping from an open interval into the collection of all Kähler metrics on M, Rg is the scalar curvature of the individual Kähler metrics, and the indices α, β correspond to arbitrary holomorphic coordinates zα.
This is a fourth-order geometric flow, as the right-hand side of the equation involves fourth derivatives of g. The Calabi flow was introduced by Eugenio Calabi in 1982 as a suggestion for the construction of extremal Kähler metrics, which were also introduced in the same paper.
It is the gradient flow of the Calabi functional; extremal Kähler metrics are the critical points of the Calabi functional.
A convergence theorem for the Calabi flow was found by Piotr Chruściel in the case that M has complex dimension equal to one.