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.
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 (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 yields a flow, and stationary states for the flow are solutions to the elliptic partial differential equation .
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