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.