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<p><b>New page</b></p><div>In the field of [[differential geometry]] in [[mathematics]], '''mean curvature flow''' is an example of a [[geometric flow]] of [[Glossary_of_differential_geometry_and_topology#H|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 [[Mathematical singularity|singularities]].<br />
<br />
Under the constraint that volume enclosed is constant, this is called [[surface tension]] flow.<br />
<br />
It is a [[parabolic partial differential equation]], and can be interpreted as "smoothing".<br />
<br />
==Physical examples==<br />
The most familiar example of mean curvature flow is in the evolution of [[soap film]]s. A similar 2 dimensional phenomenon is oil drops on the surface of water, which evolve into disks (circular boundary).<br />
<br />
Mean curvature flow was originally proposed as a model for the formation of grain boundaries in the annealing of pure metal.<br />
<br />
==Properties==<br />
The mean curvature flow extremalizes surface area, and [[minimal surface]]s are the critical points for the mean curvature flow; minima solve the [[isoperimetric]] problem.<br />
<br />
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.<br />
<br />
Related flows are:<br />
* the surface tension flow<br />
* the Lagrangian mean curvature flow<br />
* the [[inverse mean curvature flow]]<br />
<br />
==Mean Curvature Flow of a Three Dimensional Surface==<br />
The differential equation for mean-curvature flow of a surface given by <math>z=S(x,y)</math> is given by<br />
<br />
:<math>\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}<br />
</math><br />
<br />
with <math>D</math> being a constant relating the curvature and the speed of the surface normal, and<br />
the mean curvature being<br />
<br />
:<math><br />
\begin{align}<br />
H(x,y) & = <br />
\frac{1}{2}\frac{<br />
\left(1 + \left(\frac{\partial S}{\partial x}\right)^2\right) \frac{\partial^2 S}{\partial y^2} - <br />
2 \frac{\partial S}{\partial x} \frac{\partial S}{\partial y} \frac{\partial^2 S}{\partial x \partial y} + <br />
\left(1 + \left(\frac{\partial S}{\partial y}\right)^2\right) \frac{\partial^2 S}{\partial x^2}<br />
}{\left(1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2\right)^{3/2}}.<br />
\end{align}<br />
</math><br />
<br />
In the limits <math> |\frac{\partial S}{\partial x}| \ll 1 </math> and <br />
<math> |\frac{\partial S}{\partial y}| \ll 1 </math>, so that the surface is nearly planar with its normal nearly<br />
parallel to the z axis, this reduces to a [[diffusion equation]]<br />
<br />
:<math>\frac{\partial S}{\partial t} = D\ \nabla^2 S<br />
</math><br />
<br />
While the conventional diffusion equation is a linear parabolic partial differential equation and does not develop<br />
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 <br />
mean curvature flows.<br />
<br />
==References==<br />
* Ecker, Klaus. "Regularity Theory for Mean Curvature Flow", ''Progress in nonlinear differential equations and their applications'', '''75''', Birkhauser, Boston, 2004.<br />
<br />
* Mantegazza, Carlo. " Lecture Notes on Mean Curvature Flow", ''Progress in Mathematics'', '''290''', Birkhauser, Basel, 2011.<br />
<br />
* Equations 3a and 3b of C. Lu, Y. Cao, and D. Mumford. "Surface Evolution under Curvature Flows", ''Journal of Visual Communication and Image Representation'', '''13''', pp. 65-81, 2002.<br />
<br />
[[Category:Geometric flow]]<br />
[[Category:Differential geometry]]</div>en>ChrisGualtieri