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 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 $M$ be a compact smooth manifold, let $(M',g)$ be a complete smooth Riemannian manifold, and let $f:M\to M'$ be a smooth immersion. Then there is a positive number $T$ , which could be infinite, and a map $F:[0,T)\times M\to M'$ with the following properties:

$F(0,\cdot )=f$
$F(t,\cdot ):M\to M'$ is a smooth immersion for any $t\in [0,T)$
as $t\searrow 0,$ one has $F(t,\cdot )\to f$ in $C^{\infty }$
for any $(t_{0},p)\in (0,T)\times M$ , the derivative of the curve $t\mapsto F(t,p)$ at $t_{0}$ is equal to the mean curvature vector of $F(t_{0},\cdot )$ at $p$ .
if ${\widetilde {F}}:[0,{\widetilde {T}})\times M\to M'$ is any other map with the four properties above, then ${\widetilde {T}}\leq T$ and ${\widetilde {F}}(t,p)=F(t,p)$ for any $(t,p)\in [0,{\widetilde {T}})\times M.$

Necessarily, the restriction of $F$ to $(0,T)\times M$ is $C^{\infty }$ .

One refers to $F$ as the (maximally extended) mean curvature flow with initial data $f$ .

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 $(M',g)$ is the Euclidean space $\mathbb {R} ^{n+1}$ , where $n\geq 2$ denotes the dimension of $M$ , then $T$ is necessarily finite. If the second fundamental form of the 'initial immersion' $f$ is strictly positive, then the second fundamental form of the immersion $F(t,\cdot )$ is also strictly positive for every $t\in (0,T)$ , and furthermore if one choose the function $c:(0,T)\to (0,\infty )$ such that the volume of the Riemannian manifold $(M,(c(t)F(t,\cdot ))^{\ast }g_{\text{Euc}})$ is independent of $t$ , then as $t\nearrow T$ the immersions $c(t)F(t,\cdot ):M\to \mathbb {R} ^{n+1}$ smoothly converge to an immersion whose image in $\mathbb {R} ^{n+1}$ is a round sphere.

Note that if $n\geq 2$ and $f:M\to \mathbb {R} ^{n+1}$ is a smooth hypersurface immersion whose second fundamental form is positive, then the Gauss map $\nu :M\to S^{n}$ is a diffeomorphism, and so one knows from the start that $M$ is diffeomorphic to $S^{n}$ and, from elementary differential topology, that all immersions considered above are embeddings.

Gage and Hamilton extended Huisken's result to the case $n=1$ . Matthew Grayson (1987) showed that if $f:S^{1}\to \mathbb {R} ^{2}$ is any smooth embedding, then the mean curvature flow with initial data $f$ eventually consists exclusively of embeddings with strictly positive curvature, at which point Gage and Hamilton's result applies.^[4] In summary:

If $f:S^{1}\to \mathbb {R} ^{2}$ is a smooth embedding, then consider the mean curvature flow $F:[0,T)\times S^{1}\to \mathbb {R} ^{2}$ with initial data $f$ . Then $F(t,\cdot ):S^{1}\to \mathbb {R} ^{2}$ is a smooth embedding for every $t\in (0,T)$ and there exists $t_{0}\in (0,T)$ such that $F(t,\cdot ):S^{1}\to \mathbb {R} ^{2}$ has positive (extrinsic) curvature for every $t\in (t_{0},T)$ . If one selects the function $c$ as in Huisken's result, then as $t\nearrow T$ the embeddings $c(t)F(t,\cdot ):S^{1}\to \mathbb {R} ^{2}$ 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:

Curve-shortening flow, 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 $z=S(x,y)$ is given by

{\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}}}

with $D$ being a constant relating the curvature and the speed of the surface normal, and the mean curvature being

{\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}}

In the limits $\left|{\frac {\partial S}{\partial x}}\right|\ll 1$ and $\left|{\frac {\partial S}{\partial y}}\right|\ll 1$ , so that the surface is nearly planar with its normal nearly parallel to the z axis, this reduces to a diffusion equation

{\frac {\partial S}{\partial t}}=D\ \nabla ^{2}S

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 $\mathbb {R} ^{m+1}$ . The mean curvature of an $m$ -dimensional sphere of radius $R$ is $H=m/R$ .

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 $\partial _{t}F=-H\nu$ reduces to the ordinary differential equation, for an initial sphere of radius $R_{0}$ ,

{\begin{aligned}{\frac {\text{d}}{{\text{d}}t}}R(t)&=-{\frac {m}{R(t)}},\\R(0)&=R_{0}.\end{aligned}}

The solution of this ODE (obtained, e.g., by separation of variables) is

R(t)={\sqrt {R_{0}^{2}-2mt}}

,

which exists for $t\in (-\infty ,R_{0}^{2}/2m)$ .^[7]

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.

[1] 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.

[2] Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature". Journal of Differential Geometry. 17 (2): 255–306. doi:10.4310/jdg/1214436922.

[3] Huisken, Gerhard (1984). "Flow by mean curvature of convex surfaces into spheres". J. Differential Geom. 20 (1): 237–266. doi:10.4310/jdg/1214438998.

[4] 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.

[5] 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.

[6] 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.

[7] 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.

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