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 or a parameter space.
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.
Intrinsic geometric flows are flows on the Riemannian metric, independent of any embedding or immersion.
Important classes of flows are curvature flows, variational flows, 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 yields a flow, and stationary states for the flow are solutions to the elliptic partial differential equation.
If the equation 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, which has Euler–Lagrange equation for some elliptic operator L, and associated parabolic PDE.
The Ricci flow, Calabi flow, and Yamabe flow arise in this way.
Curvature flows may or may not preserve volume, 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.