Calabi flow

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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.

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 extremalizes a certain intrinsic curvature functional. 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.

[edit] References

• Bakas, I.. The algebraic structure of geometric flows in two dimensions. arXiv eprint server. Retrieved on July 28, 2005.