Calabi flow

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:

[math]\displaystyle{ \frac{\partial g_{\alpha\overline{\beta}}}{\partial t}=\frac{\partial^2 R^g}{\partial z^\alpha\partial\overline{z}^\beta} }[/math],

where g is a mapping from an open interval into the collection of all Kähler metrics on M, R^g is the scalar curvature of the individual Kähler metrics, and the indices α, β correspond to arbitrary holomorphic coordinates z^α. 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. 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 (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.