Mean curvature flow
Encyclopedia
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 surfaces evolves under mean curvature flow if the velocity of which a point on the surface moves is given by the mean curvature
of the surface. For example, a round sphere
evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities
.
Under the constraint that volume enclosed is constant, this is called surface tension
flow.
It is a parabolic partial differential equation
, and can be interpreted as "smoothing".
s.
A similar 2 dimensional phenomenon is oil drops on the surface of water, which evolve into disks (circular boundary).
s are the critical points for the mean curvature flow; minima solve the isoperimetric problem.
For manifolds embedded in a symplectic manifold, if the surface is a Lagrangian submanifold, the mean curvature flow is of Lagrangian type, so the surface evolves within the class of Lagrangian submanifolds.
Related flows are:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, mean curvature flow is an example of a geometric flow
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...
of hypersurfaces in a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
(for example, smooth surfaces in 3-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
). Intuitively, a family of surfaces evolves under mean curvature flow if the velocity of which a point on the surface moves is given by the mean curvature
Mean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....
of the surface. For example, a round sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
.
Under the constraint that volume enclosed is constant, this is called surface tension
Surface tension
Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...
flow.
It is a parabolic partial differential equation
Parabolic partial differential equation
A parabolic partial differential equation is a type of second-order partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
, and can be interpreted as "smoothing".
Physical examples
The most familiar example of mean curvature flow is in the evolution of soap filmSoap film
Soap films are thin layers of liquid surrounded by air. For example, if two soap bubbles enters in contact, they merged and a thin film is created in between. Thus, foams are composed of a network of films connected by Plateau borders...
s.
A similar 2 dimensional phenomenon is oil drops on the surface of water, which evolve into disks (circular boundary).
Properties
The mean curvature flow extremalizes surface area, and minimal surfaceMinimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s are the critical points for the mean curvature flow; minima solve the isoperimetric problem.
For manifolds embedded in a symplectic manifold, if the surface is a Lagrangian submanifold, the mean curvature flow is of Lagrangian type, so the surface evolves within the class of Lagrangian submanifolds.
Related flows are:
- the surface tension flow
- the Lagrangian mean curvature flow
- the inverse mean curvature flowInverse mean curvature flowIn the field of differential geometry in mathematics, inverse mean curvature flow is an example of a geometric flow of hypersurfaces a Riemannian manifold...