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
• Curve-shortening flow, the one-dimensional case of the mean curvature flow
• Willmore flow, as in minimax eversions of spheres
• Inverse mean curvature flow

Intrinsic

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

• Ricci flow, as in the solution of the Poincaré conjecture, and Richard Hamilton's proof of the uniformization theorem
• Calabi flow, two dimensional and for string theory
• Yamabe flow, be some special case of the Ricci flow

Classes of flows

Important classes of flows are curvature flows, variational flows (which extremize some functional), and flows arising as solutions to parabolic partial differential equations. A given flow frequently admits all of these interpretations, as follows.

Given an elliptic operator L, the parabolic PDE $u_t = Lu$ yields a flow, and stationary states for the flow are solutions to the elliptic partial differential equation $Lu=0$.

If the equation $Lu=0$ is the Euler–Lagrange equation for some functional F, then the flow has a variational interpretation as the gradient flow of F, and stationary states of the flow correspond to critical points of the functional.

In the context of geometric flows, the functional is often the L² norm of some curvature.

Thus, given a curvature K, one can define the functional $F(K)=\|K\|_2 := \left(\int_M K^2\right)^{1/2}$, which has Euler–Lagrange equation $Lu=0$ for some elliptic operator L, and associated parabolic PDE $u_t=Lu$.

The Ricci flow, Calabi flow, and Yamabe flow arise in this way (in some cases with normalizations).

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, Ioannis (14 October 2005) [28 Jul 2005 (v1)]. "The algebraic structure of geometric flows in two dimensions". Journal of High Energy Physics 2005 (10): 038. arXiv:hep-th/0507284. doi:10.1088/1126-6708/2005/10/038.

• Bakas, Ioannis (5 Feb 2007). "Renormalization group equations and geometric flows". arXiv:hep-th/0702034.