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. Let M {\displaystyle M} be a compact smooth manifold, let ( M ′ , g ) {\displaystyle (M',g)} be a complete smooth Riemannian manifold, and let f : M → M ′ {\displaystyle f:M\to M'} be a smooth immersion. Then there is a positive number T {\displaystyle T} , which could be infinite, and a map F : [ 0 , T ) × M → M ′ {\displaystyle F:[0,T)\times M\to M'} with the following properties:
F ( 0 , ⋅ ) = f {\displaystyle F(0,\cdot )=f}
F ( t , ⋅ ) : M → M ′ {\displaystyle F(t,\cdot ):M\to M'} is a smooth immersion for any t ∈ [ 0 , T ) {\displaystyle t\in [0,T)}
as t ↘ 0 , {\displaystyle t\searrow 0,} one has F ( t , ⋅ ) → f {\displaystyle F(t,\cdot )\to f} in C ∞ {\displaystyle C^{\infty }}
for any ( t 0 , p ) ∈ ( 0 , T ) × M {\displaystyle (t_{0},p)\in (0,T)\times M} , the derivative of the curve t ↦ F ( t , p ) {\displaystyle t\mapsto F(t,p)} at t 0 {\displaystyle t_{0}} is equal to the mean curvature vector of F ( t 0 , ⋅ ) {\displaystyle F(t_{0},\cdot )} at p {\displaystyle p} . if F ~ : [ 0 , T ~ ) × M → M ′ {\displaystyle {\widetilde {F}}:[0,{\widetilde {T}})\times M\to M'} is any other map with the four properties above, then T ~ ≤ T {\displaystyle {\widetilde {T}}\leq T} and F ~ ( t , p ) = F ( t , p ) {\displaystyle {\widetilde {F}}(t,p)=F(t,p)} for any ( t , p ) ∈ [ 0 , T ~ ) × M . {\displaystyle (t,p)\in [0,{\widetilde {T}})\times M.}
Necessarily, the restriction of F {\displaystyle F} to ( 0 , T ) × M {\displaystyle (0,T)\times M} is C ∞ {\displaystyle C^{\infty }} . One refers to F {\displaystyle F} as the (maximally extended) mean curvature flow with initial data f {\displaystyle 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:
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