In the mathematical field of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, a geometric flow, also called a geometric evolution equation, is a type of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

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 equation A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivati ...

s. Certain geometric flows arise as the gradient flow associated to a functional on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature. Such flows are fundamentally related to the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, and include

mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surf ...

and

Yamabe flow In differential geometry, the Yamabe flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold. First introduced by Richard S. Hamilton, Yamabe flow is for noncompact manifolds, and is the negative ''L'' ...

Examples

Extrinsic

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

Mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surf ...

, as in soap films; critical points are minimal surfaces * Curve-shortening flow, the one-dimensional case of the mean curvature flow * Willmore flow, as in minimax eversions 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 Poincaré conjecture Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...

, and

Richard S. Hamilton Richard Streit Hamilton (born 10 January 1943) is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. Hamilton ...

's proof of the uniformization theorem *

Calabi flow In the mathematical fields of differential geometry and geometric analysis, the Calabi flow is a geometric flow which deforms a Kähler metric on a complex manifold. Precisely, given a Kähler manifold , the Calabi flow is given by: :\frac=\frac, wh ...

, a flow for Kähler metrics *

Classes of flows

Important classes of flows are curvature flows, variational flows (which extremize some functional), and flows arising as solutions to

s. A given flow frequently admits all of these interpretations, as follows. Given an

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which i ...

L,

the parabolic PDE

u_t = Lu

yields a flow, and stationary states for the flow are solutions to the

elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, wher ...

Lu = 0.

If the equation

Lu = 0

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

F(K) = \, K\, _2 := \left(\int_M K^2\right)^.

which has Euler–Lagrange equation

Lu=0

for some elliptic operator

L,

and associated parabolic PDE

u_t = Lu.

The Ricci flow,

, and

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

* * {{DEFAULTSORT:Geometric Flow

Examples

Extrinsic

Intrinsic

Classes of flows

See also

References