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Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature. (English) Zbl 0804.53006
The authors study the mean curvature flow of axially symmetric hypersurfaces in $$\mathbb{R}^ n$$ $$(n\geq 3)$$. Let $$\Gamma_ 0$$ be a smooth initial surface parametrized by $$r= h_ 0(x_ 1)$$, where $$r= \sqrt{x^ 2_ 2+\cdots+ x^ 2_ n}$$. Due to Evans and Spruck there exists a unique viscosity solution of the motion by mean curvature for $$\Gamma_ 0$$, called $$\Gamma_ t$$ $$(t\geq 0)$$. As long as $$\Gamma_ t$$ is smooth it can be parametrized by $$r= h(x_ 1,t)$$ which solves the nonlinear parabolic equation $$h_ t- {h_{x_ 1 x_ 1}\over 1+ h^ 2_{x_ 1}}+ {n-2\over h}= 0$$ with initial data $$h(\cdot,0)= h_ 0$$.
The main result is that $$\lim_{t\uparrow T} (T- t)^{-1/2} h(y(T- t)^{1/2},t)= \sqrt{2(n-2)}$$ if $$h$$ vanishes for the first time at $$(0,T)$$, $$h(x_ 1, t)= h(- x_ 1,t)$$ and if $$x_ 1 h_{x_ 1}(x_ 1,t)\geq 0$$.
As consequences of this one has that up to parabolic scaling, at the singularity the surface $$\Gamma_ t$$ converges to a cylinder, $$\Gamma_ t$$ does not develop an interior and continues as a smooth surface after the singularity until extinction time.

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 37C10 Dynamics induced by flows and semiflows 35K55 Nonlinear parabolic equations
##### Keywords:
mean curvature flow; singularity; cylinder
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##### References:
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