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