In the mathematical fields of
differential geometry 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
In mathem ...
, the Calabi flow is a
geometric flow In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding. It is not a term with ...
which deforms a
Kähler metric Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
* Erich Kähler (1906–2000), German mathematician
* Heinz Kähler (1905–1974), German art historian and ...
on a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
. 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 Ar ...
, the Calabi flow is given by:
:
,
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 derivatives of .
The Calabi flow was introduced by
Eugenio Calabi
Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...
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 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 Univ. Press, 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