In the 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 multilin ...
in
mathematics, mean curvature flow is an example of a
geometric flow of
hypersurfaces in a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
(for example, smooth surfaces in 3-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the
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.
The ...
of the surface. For example, a round
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
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
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to ...
flow.
It is a
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 ...
, and can be interpreted as "smoothing".
Existence and uniqueness
The following was shown by
Michael Gage and
Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows.
Let
be a compact
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, let
be a complete smooth
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
, and let
be a smooth
immersion. Then there is a positive number
, which could be infinite, and a map
with the following properties:
*
*
is a smooth immersion for any
* as
one has
in
* for any
, the derivative of the curve
at
is equal to the
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.
The ...
vector of
at
.
* if
is any other map with the four properties above, then
and
for any
Necessarily, the restriction of
to
is
.
One refers to
as the (maximally extended) mean curvature flow with initial data
.
Convergence theorems
Following Hamilton's epochal 1982 work on the Ricci flow, in 1984
Gerhard Huisken
Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huiske ...
employed the same methods for the mean curvature flow to produce the following analogous result:
* If
is the Euclidean space
, where
denotes the dimension of
, then
is necessarily finite. If the second fundamental form of the 'initial immersion'
is strictly positive, then the second fundamental form of the immersion
is also strictly positive for every
, and furthermore if one choose the function
such that the volume of the Riemannian manifold
is independent of
, then as
the immersions
smoothly converge to an immersion whose image in
is a round sphere.
Note that if
and
is a smooth hypersurface immersion whose second fundamental form is positive, then the
Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that '' ...
is a diffeomorphism, and so one knows from the start that
is diffeomorphic to
and, from elementary differential topology, that all immersions considered above are embeddings.
Gage and Hamilton extended Huisken's result to the case
. Matthew Grayson (1987) showed that if
is any smooth embedding, then the mean curvature flow with initial data
eventually consists exclusively of embeddings with strictly positive curvature, at which point Gage and Hamilton's result applies. In summary:
* If
is a smooth embedding, then consider the mean curvature flow
with initial data
. Then
is a smooth embedding for every
and there exists
such that
has positive (extrinsic) curvature for every
. If one selects the function
as in Huisken's result, then as
the embeddings
converge smoothly to an embedding whose image is a round circle.
Physical examples
The most familiar example of mean curvature flow is in the evolution of soap films. A similar 2-dimensional phenomenon is oil drops on the surface of water, which evolve into disks (circular boundary).
Mean curvature flow was originally proposed as a model for the formation of grain boundaries in the annealing of pure metal.
Properties
The mean curvature flow
extremalizes surface area, and
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s are the critical points for the mean curvature flow; minima solve the
isoperimetric problem.
For manifolds embedded in a
Kähler–Einstein 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.
Huisken's monotonicity formula gives a monotonicity property of the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of a time-reversed
heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum o ...
with a surface undergoing the mean curvature flow.
Related flows are:
*
Curve-shortening flow
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a g ...
, the one-dimensional case of mean curvature flow
* the surface tension flow
* the Lagrangian mean curvature flow
* the
inverse mean curvature flow
Mean curvature flow of a three-dimensional surface
The differential equation for mean-curvature flow of a surface given by
is given by
:
with
being a constant relating the curvature and the speed of the surface normal, and
the mean curvature being
:
In the limits
and
, so that the surface is nearly planar with its normal nearly
parallel to the z axis, this reduces to a
diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws ...
:
While the conventional diffusion equation is a linear parabolic partial differential equation and does not develop
singularities (when run forward in time), mean curvature flow may develop singularities because it is a nonlinear parabolic equation. In general additional constraints need to be put on a surface to prevent singularities under
mean curvature flows.
Every smooth convex surface collapses to a point under the mean-curvature flow, without other singularities, and converges to the shape of a sphere as it does so. For surfaces of dimension two or more this is a theorem of
Gerhard Huisken
Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huiske ...
; for the one-dimensional
curve-shortening flow
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a g ...
it is the Gage–Hamilton–Grayson theorem. However, there exist embedded surfaces of two or more dimensions other than the sphere that stay self-similar as they contract to a point under the mean-curvature flow, including the
Angenent torus.
Example: mean curvature flow of ''m''-dimensional spheres
A simple example of mean curvature flow is given by a family of concentric round
hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...
s in
. The mean curvature of an
-dimensional sphere of radius
is
.
Due to the rotational symmetry of the sphere (or in general, due to the invariance of mean curvature under
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
) the mean curvature flow equation
reduces to the
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
, for an initial sphere of radius
,
:
The solution of this ODE (obtained, e.g., by
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
) is
:
,
which exists for
.
[.]
References
*.
*.
*{{citation
, last1 = Lu , first1 = Conglin
, last2 = Cao , first2 = Yan
, last3 = Mumford , first3 = David , author3-link = David Mumford
, doi = 10.1006/jvci.2001.0476
, issue = 1–2
, journal =
Journal of Visual Communication and Image Representation
The ''Journal of Visual Communication and Image Representation'' is a peer-reviewed academic journal of media studies published by Elsevier. It was established in 1990 and is published in 8 issues per year. The editors-in-chief are M.T. Sun (Univer ...
, pages = 65–81
, title = Surface evolution under curvature flows
, volume = 13
, year = 2002, s2cid = 7341932
. See in particular Equations 3a and 3b.
Geometric flow
Differential geometry