# Geometric flow

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In the [mathematical](/source/mathematics) field of [differential geometry](/source/differential_geometry), a '''geometric flow''', also called a '''geometric evolution equation''', is a type of [partial differential equation](/source/partial_differential_equation) for a geometric object such as a [Riemannian metric](/source/Riemannian_metric) or an [embedding](/source/embedded_submanifold). It is not a term with a formal meaning, but is typically understood to refer to [parabolic partial differential equation](/source/parabolic_partial_differential_equation)s.

Certain geometric flows arise as the [gradient flow](/source/gradient_flow) associated with a functional on a [manifold](/source/manifold) which has a geometric interpretation, usually associated with some [extrinsic or intrinsic curvature](/source/Curvature). Such flows are fundamentally related to the [calculus of variations](/source/calculus_of_variations), and include [mean curvature flow](/source/mean_curvature_flow) and [Yamabe flow](/source/Yamabe_flow).

==Examples==
===Extrinsic===
Extrinsic geometric flows are flows on [embedded submanifold](/source/embedded_submanifold)s, or more generally
[immersed submanifold](/source/immersed_submanifold)s. In general they change both the Riemannian metric and the immersion.
* [Mean curvature flow](/source/Mean_curvature_flow), as in [soap film](/source/soap_film)s; critical points are [minimal surface](/source/minimal_surface)s
* [Curve-shortening flow](/source/Curve-shortening_flow), the one-dimensional case of the mean curvature flow
* [Willmore flow](/source/Willmore_flow), as in [minimax eversion](/source/minimax_eversion)s of spheres
* [Inverse mean curvature flow](/source/Inverse_mean_curvature_flow)

===Intrinsic===
Intrinsic geometric flows are flows on the [Riemannian metric](/source/Riemannian_metric), independent of any embedding or immersion.
* [Ricci flow](/source/Ricci_flow), as in the [solution of the Poincaré conjecture](/source/solution_of_the_Poincar%C3%A9_conjecture), and [Richard S. Hamilton](/source/Richard_S._Hamilton)'s proof of the [uniformization theorem](/source/uniformization_theorem)
* [Calabi flow](/source/Calabi_flow), a flow for [Kähler metric](/source/K%C3%A4hler_metric)s
* [Yamabe flow](/source/Yamabe_flow)

==Classes of flows==
Important classes of flows are '''curvature flows''', '''variational flows''' (which extremize some functional), and flows arising as solutions to [parabolic partial differential equation](/source/parabolic_partial_differential_equation)s. A given flow frequently admits all of these interpretations, as follows.

Given an [elliptic operator](/source/elliptic_operator) <math>L,</math> the parabolic PDE <math>u_t = Lu</math> yields a flow, and stationary states for the flow are solutions to the [elliptic partial differential equation](/source/elliptic_partial_differential_equation) <math>Lu = 0.</math>

If the equation <math>Lu = 0</math> is the [Euler–Lagrange equation](/source/Euler%E2%80%93Lagrange_equation) for some functional <math>F,</math> then the flow has a variational interpretation as the gradient flow of <math>F,</math> and stationary states of the flow correspond to critical points of the functional.

In the context of geometric flows, the functional is often the [<math>L^2</math> norm](/source/L2_norm) of some curvature.

Thus, given a curvature <math>K,</math> one can define the functional <math>F(K) = \|K\|_2 := \left(\int_M K^2\right)^{1/2},</math> which has Euler–Lagrange equation <math>Lu=0</math> for some elliptic operator <math>L,</math> and associated parabolic PDE <math>u_t = Lu.</math>

The [Ricci flow](/source/Ricci_flow), [Calabi flow](/source/Calabi_flow), and [Yamabe flow](/source/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 also==
* [Harmonic map heat flow](/source/Harmonic_map)
* [Curve-shortening flow](/source/Curve-shortening_flow)

==References==

{{reflist}}
{{reflist|group=note}}

* {{cite journal
| last = Bakas | first = Ioannis
| title = The algebraic structure of geometric flows in two dimensions
| orig-year = 28 Jul 2005 (v1)
| arxiv = hep-th/0507284
| journal = [Journal of High Energy Physics](/source/Journal_of_High_Energy_Physics)
| volume = 2005 | issue = 10 | page = 038| date = 14 October 2005
| doi = 10.1088/1126-6708/2005/10/038
| bibcode = 2005JHEP...10..038B|s2cid = 15924056
}}

* {{cite arXiv
| last = Bakas | first = Ioannis
| title = Renormalization group equations and geometric flows
| date = 2007
| eprint = hep-th/0702034
}}

{{Manifolds}}

{{DEFAULTSORT:Geometric Flow}}
Category:Geometric flow

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Adapted from the Wikipedia article [Geometric flow](https://en.wikipedia.org/wiki/Geometric_flow) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Geometric_flow?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.