Geometric flow

Summary

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 to 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.

ExamplesEdit

ExtrinsicEdit

Extrinsic geometric flows are flows on embedded submanifolds, or more generally immersed submanifolds. In general they change both the Riemannian metric and the immersion.

IntrinsicEdit

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

Classes of flowsEdit

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

If the equation   is the Euler–Lagrange equation for some functional   then the flow has a variational interpretation as the gradient flow of   and stationary states of the flow correspond to critical points of the functional.

In the context of geometric flows, the functional is often the   norm norm of some curvature.

Thus, given a curvature   one can define the functional   which has Euler–Lagrange equation   for some elliptic operator   and associated parabolic PDE  

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.

See alsoEdit

ReferencesEdit

  • Bakas, Ioannis (14 October 2005) [28 Jul 2005 (v1)]. "The algebraic structure of geometric flows in two dimensions". Journal of High Energy Physics. 2005 (10): 038. arXiv:hep-th/0507284. Bibcode:2005JHEP...10..038B. doi:10.1088/1126-6708/2005/10/038. S2CID 15924056.
  • Bakas, Ioannis (5 Feb 2007). "Renormalization group equations and geometric flows". arXiv:hep-th/0702034. Bibcode:2007hep.th....2034B. {{cite journal}}: Cite journal requires |journal= (help)