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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]].

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".

==Physical examples==
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).

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]]s are the critical points for the mean curvature flow; minima solve the [[isoperimetric]] problem.

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.

Related flows are:
* 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 <math>z=S(x,y)</math> is given by

:<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}
</math>

with <math>D</math> being a constant relating the curvature and the speed of the surface normal, and
the mean curvature being

:<math>
\begin{align}
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{align}
</math>

In the limits <math> |\frac{\partial S}{\partial x}| \ll 1 </math> and
<math> |\frac{\partial S}{\partial y}| \ll 1 </math>, so that the surface is nearly planar with its normal nearly
parallel to the z axis, this reduces to a [[diffusion equation]]

:<math>\frac{\partial S}{\partial t} = D\ \nabla^2 S
</math>

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.

==References==
* Ecker, Klaus. "Regularity Theory for Mean Curvature Flow", ''Progress in nonlinear differential equations and their applications'', '''75''', Birkhauser, Boston, 2004.

* Mantegazza, Carlo. " Lecture Notes on Mean Curvature Flow", ''Progress in Mathematics'', '''290''', Birkhauser, Basel, 2011.

* 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.

[[Category:Geometric flow]]
[[Category:Differential geometry]]

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