- Geometric flow
In
mathematics , specificallydifferential geometry , a geometric flow is thegradient flow associated to a functional on amanifold which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature. They can be interpreted as flows on amoduli space (for intrinsic flows) or aparameter space (for extrinsic flows).These are of fundamental interest in the
calculus of variations , and include several famous problems and theories.Particularly interesting are theircritical 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 generallyimmersed submanifold s. In general they change both the Riemannian metric and the immersion.
*Mean curvature flow , as insoap film s; critical points areminimal surface s
*Willmore flow , as inminimax 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 theSolution of the Poincaré conjecture , and Richard Hamilton's proof of theUniformization 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 yields a flow, and stationary states for the flow are solutions to theelliptic 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 "L"2 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 , andYamabe 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.
References
* cite journal
author=Bakas, I.
title=The algebraic structure of geometric flows in two dimensions
year=2005
id=arxiv|hep-th|0507284
accessdate=July 28
accessyear=2005* cite journal
author=Bakas, I.
title=Renormalization group equations and geometric flows
year=2007
id=arxiv|hep-th|0702034
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