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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.

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
37C10 Dynamics induced by flows and semiflows
35K55 Nonlinear parabolic equations
Full Text: DOI
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