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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 point] s.

A geometric flow is also called a geometric evolution equation .

Examples
Extrinsic
Extrinsic geometric flows are flows on [embedded submanifold] s, or more generally
[immersed submanifold] s. In general they change both the Riemannian metric and the immersion.
- [Mean curvature flow] , as in [soap film] s; critical points are [minimal surface] s
- [Willmore flow] , as in [minimax eversion] s 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 PoincarĂ© conjecture] , and [Richard Hamilton's] proof of the [Uniformization theorem]
- [Calabi flow]
- [Yamabe flow]

Classes of flows
Important classes of flows are curvature flows , variational flows (which extremelize some functional), and flows arising as solutions to [parabolic partial differential equation] s. 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.

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