In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding.
It is not a term with a formal meaning, but is typically understood to refer to parabolic partial differential equations.
Certain geometric flows arise as the gradient flow associated with a functional on a manifold which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature.
Such flows are fundamentally related to the calculus of variations, and include mean curvature flow and Yamabe flow.
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.
yields a flow, and stationary states for the flow are solutions to the elliptic partial differential equation
and stationary states of the flow correspond to critical points of the functional.
In the context of geometric flows, the functional is often the
Thus one often normalizes the flow, for instance, by fixing the volume.