Parabolic partial differential equation
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 normal component of 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".
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.[1][2]
Let
be a compact smooth manifold, let
be a complete smooth Riemannian manifold, and let
be a smooth immersion. Then there is a positive number
, which could be infinite, and a map
with the following properties:
![{\displaystyle F(0,\cdot )=f}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is a smooth immersion for any ![{\displaystyle t\in [0,T)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- as
one has
in ![{\displaystyle C^{\infty }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- for any
, the derivative of the curve
at
is equal to the mean curvature vector of
at
. - if
is any other map with the four properties above, then
and
for any ![{\displaystyle (t,p)\in [0,{\widetilde {T}})\times M.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Necessarily, the restriction of
to
is
.
One refers to
as the (maximally extended) mean curvature flow with initial data
.
Convex solutions
Following Hamilton's epochal 1982 work on the Ricci flow, in 1984 Gerhard Huisken employed the same methods for the mean curvature flow to produce the following analogous result:[3]
- 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
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.[4] 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.
Properties
The mean curvature flow extremalizes surface area, and minimal surfaces 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 of a time-reversed heat kernel with a surface undergoing the mean curvature flow.
Related flows are:
Mean curvature flow of a three-dimensional surface
The differential equation for mean-curvature flow of a surface given by
is given by
![{\displaystyle {\frac {\partial S}{\partial t}}=2D\ H(x,y){\sqrt {1+\left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
being a constant relating the curvature and the speed of the surface normal, and
the mean curvature being
![{\displaystyle {\begin{aligned}H(x,y)&={\frac {1}{2}}{\frac {\left(1+\left({\frac {\partial S}{\partial x}}\right)^{2}\right){\frac {\partial ^{2}S}{\partial y^{2}}}-2{\frac {\partial S}{\partial x}}{\frac {\partial S}{\partial y}}{\frac {\partial ^{2}S}{\partial x\partial y}}+\left(1+\left({\frac {\partial S}{\partial y}}\right)^{2}\right){\frac {\partial ^{2}S}{\partial x^{2}}}}{\left(1+\left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}\right)^{3/2}}}.\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
![{\displaystyle {\frac {\partial S}{\partial t}}=D\ \nabla ^{2}S}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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;[5] for the one-dimensional curve-shortening flow 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.[6]
Example: mean curvature flow of m-dimensional spheres
A simple example of mean curvature flow is given by a family of concentric round hyperspheres 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) the mean curvature flow equation
reduces to the ordinary differential equation, for an initial sphere of radius
,
![{\displaystyle {\begin{aligned}{\frac {\text{d}}{{\text{d}}t}}R(t)&=-{\frac {m}{R(t)}},\\R(0)&=R_{0}.\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The solution of this ODE (obtained, e.g., by separation of variables) is
,
which exists for
.[7]
See Also
References
- ^ Gage, M.; Hamilton, R.S. (1986). "The heat equation shrinking convex plane curves". J. Differential Geom. 23 (1): 69–96. doi:10.4310/jdg/1214439902.
- ^ Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature". Journal of Differential Geometry. 17 (2): 255–306. doi:10.4310/jdg/1214436922.
- ^ Huisken, Gerhard (1984). "Flow by mean curvature of convex surfaces into spheres". J. Differential Geom. 20 (1): 237–266. doi:10.4310/jdg/1214438998.
- ^ Grayson, Matthew A. (1987). "The heat equation shrinks embedded plane curves to round points". J. Differential Geom. 26 (2): 285–314. doi:10.4310/jdg/1214441371.
- ^ Huisken, Gerhard (1990), "Asymptotic behavior for singularities of the mean curvature flow", Journal of Differential Geometry, 31 (1): 285–299, doi:10.4310/jdg/1214444099, hdl:11858/00-001M-0000-0013-5CFD-5, MR 1030675.
- ^ Angenent, Sigurd B. (1992), "Shrinking doughnuts" (PDF), Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), Progress in Nonlinear Differential Equations and their Applications, vol. 7, Boston, MA: Birkhäuser, pp. 21–38, MR 1167827.
- ^ Ecker, Klaus (2004), Regularity Theory for Mean Curvature Flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Boston, MA: Birkhäuser, doi:10.1007/978-0-8176-8210-1, ISBN 0-8176-3243-3, MR 2024995.
- Ecker, Klaus (2004), Regularity Theory for Mean Curvature Flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Boston, MA: Birkhäuser, doi:10.1007/978-0-8176-8210-1, ISBN 0-8176-3243-3, MR 2024995.
- Mantegazza, Carlo (2011), Lecture Notes on Mean Curvature Flow, Progress in Mathematics, vol. 290, Basel: Birkhäuser/Springer, doi:10.1007/978-3-0348-0145-4, ISBN 978-3-0348-0144-7, MR 2815949.
- Lu, Conglin; Cao, Yan; Mumford, David (2002), "Surface evolution under curvature flows", Journal of Visual Communication and Image Representation, 13 (1–2): 65–81, CiteSeerX 10.1.1.679.6535, doi:10.1006/jvci.2001.0476, S2CID 7341932. See in particular Equations 3a and 3b.