In the mathematical fields of

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...

, the Calabi flow is a geometric flow which deforms a Kähler metric on a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

. Precisely, given a

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...

, the Calabi flow is given by: :

\frac=\frac

, where is a mapping from an open interval into the collection of all Kähler metrics on , is the

scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...

of the individual Kähler metrics, and the indices correspond to arbitrary holomorphic coordinates . This is a fourth-order geometric flow, as the right-hand side of the equation involves fourth

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

s of . 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 '; 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 has complex dimension equal to one. 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. 102 (1982), pp. 259–290. Seminar on Differential Geometry.

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

(PUP), Princeton, N.J. * E. Calabi and X.X. Chen. The space of Kähler metrics. II. 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 Geometric flow Partial differential equations String theory