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

Types of flows

Many variants of the Ricci flow have also been studied:

• Various 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,
• Various flows which extremalize some quantity mathematically analogous to an energy or entropy,
• Various flows controlled by a p.d.e. which is a higher order analog of a nonlinear diffusion equation.

Some of the most interesting variants are examples of all of these possibilities. In particular, the Calabi flow, which, like the Ricci flow, is an intrinsic curvature flow. This flow 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 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.

Curvature flows may or may not preserve volume. The Calabi flow does; the Ricci flow does not, so to be more careful in applying the Ricci flow to uniformization we'd need to normalize the Ricci flow to obtain a flow which preserves volume. If we fail to do this, the problem is that (for example) instead of evolving a given three-dimensional manifold into one of Thurston's canonical forms, we might just shrink its size.

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)