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Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space. (English) Zbl 0909.53045

For entire graphs defining spacelike hypersurfaces in Minkowski space which are moving by mean curvature, the author proves the following Theorem: For any smooth initial data \(u_0:{\mathbb R}^n\to{\mathbb R}\) defining a spacelike hypersurface the mean curvature flow has a smooth spacelike solution for all \(t>0\). The proof is done by solving corresponding boundary value problems on increasing domains of radius \(k\) and selecting a converging subsequence of solutions for \(k\to\infty\). The longtime existence of solutions for these boundary value problems is shown in the first part of the paper by deriving appropriate interior estimates.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
35K15 Initial value problems for second-order parabolic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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