Geometric flow

In mathematics, specifically differential geometry, a geometric flow is the gradient flow associated to a functional on a manifold which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature. They can be interpreted as flows on a moduli space (for intrinsic flows) or a parameter space (for extrinsic flows).

These are of fundamental interest in the calculus of variations, and include several famous problems and theories. Particularly interesting are their critical points.

A geometric flow is also called a geometric evolution equation.

Examples

Extrinsic

Extrinsic geometric flows are flows on embedded submanifolds, or more generally immersed submanifolds. In general they change both the Riemannian metric and the immersion.

• Mean curvature flow, as in soap films; critical points are minimal surfaces
• Willmore flow, as in minimax eversions of spheres

Intrinsic

Intrinsic geometric flows are flows on the Riemannian metric, independent of any embedding or immersion.

• Ricci flow, as in the Hamilton–Perelman solution of the Poincaré conjecture, and Richard Hamilton's proof of the Uniformization theorem
• Calabi flow
• Yamabe flow

Classes of flows

Classes of flows include:

• curvature flows defined using either an extrinsic curvature, which describes how a curve or surface is embedded in a higher dimensional flat space, or an intrinsic curvature, which describes the internal geometry of some Riemannian manifold,
• flows which extremalize some quantity mathematically analogous to an energy or entropy,
• flows controlled by a partial differential equation which is a higher order analog of a nonlinear diffusion equation.

Some of the most interesting flows are examples of all of these possibilities, such as the Ricci flow and the Calabi flow.

Curvature flows may or may not preserve volume (the Calabi flow does, while the Ricci flow does not), and if not, the flow may simply shrink or grow the manifold, rather than regularizing the metric. Thus one often normalizes the flow, for instance by fixing the volume.

References

• Bakas, I. "The algebraic structure of geometric flows in two dimensions". arXiv eprint server. Retrieved July 28. {{cite web}}: Check date values in: |accessdate= (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help)