Calabi flow

In differential geometry, the Calabi flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold—in a manner formally analogous to the way that vibrations are damped and dissipated in a hypothetical curved n-dimensional structural element.

Introduction

The Calabi flow is an intrinsic curvature flow, like the Ricci flow. It tends to smooth out deviations from roundness in a manner formally analogous to the way that the two-dimensional vibration equation damps and propagates away transverse mechanical vibrations in a thin plate, and it extremizes a certain intrinsic curvature functional.

Formal statement

If Σ is a closed Riemannian surface, then the Calabi flow is given by:[1]

${\displaystyle {\frac {\partial g_{ij}}{\partial t}}=(\Delta R)g_{ij}}$,

where the ${\displaystyle g_{ij}}$ are the coordinates of the metric, ${\displaystyle \Delta }$ is the Laplace-Beltrami operator and R is the scalar curvature.

Use

The Calabi flow is important in the study of Kähler manifolds, particularly Calabi–Yau manifolds and also in the study of Robinson–Trautman spacetimes in general relativity. An intriguing observation is that the underlying Calabi equation appears to be completely integrable, which would give a direct link with the theory of solitons.

Notes

1. S.-C. Chang The two-dimensional Calabi flow Nagoya Math. J., Vol. 181 (2006), 63–73

References

• Bakas, I., The algebraic structure of geometric flows in two dimensions, arXiv:hep-th/0507284, Bibcode:2005JHEP...10..038B