How Continuous Are Viscosity Solutions to Mean Curvature Flow

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 [edit]

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

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

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

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

Convergence theorems [edit]

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

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

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

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

Physical examples [edit]

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 [edit]

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 [edit]

The differential equation for mean-curvature flow of a surface given by ${\displaystyle z=S(x,y)}$ is given by

${\displaystyle {\frac {\partial S}{\partial t}}=2D\ H(x,y){\sqrt {1+\left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}}}}$

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

${\displaystyle {\begin{aligned}H(x,y)&={\frac {1}{2}}{\frac {\left(1+\left({\frac {\partial S}{\partial x}}\right)^{2}\right){\frac {\partial ^{2}S}{\partial y^{2}}}-2{\frac {\partial S}{\partial x}}{\frac {\partial S}{\partial y}}{\frac {\partial ^{2}S}{\partial x\partial y}}+\left(1+\left({\frac {\partial S}{\partial y}}\right)^{2}\right){\frac {\partial ^{2}S}{\partial x^{2}}}}{\left(1+\left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}\right)^{3/2}}}.\end{aligned}}}$

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

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

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 [edit]

A simple example of mean curvature flow is given by a family of concentric round hyperspheres in ${\displaystyle \mathbb {R} ^{m+1}}$ . The mean curvature of an ${\displaystyle m}$ -dimensional sphere of radius ${\displaystyle R}$ is ${\displaystyle H=m/R}$ .

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

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

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

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

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

References [edit]

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, ISBN0-8176-3243-3, 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, ISBN0-8176-3243-3, 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, ISBN978-3-0348-0144-7, 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, doi:10.1006/jvci.2001.0476, S2CID 7341932 . See in particular Equations 3a and 3b.

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Source: https://en.wikipedia.org/wiki/Mean_curvature_flow

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