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Interior estimates for hypersurfaces moving by mean curvature. (English) Zbl 0707.53008
We prove local bounds on gradient, curvature and higher order geometric quantities for hypersurfaces in $${\mathbb{R}}^{n+1}$$ moving by mean curvature. Furthermore a shorttime existence result for a large class of noncompact hypersurfaces is derived as well as a maximum principle for heat equations on complete manifolds with time dependent metric. A major application of the interior estimates is the result that mean curvature flow admits a smooth solution for all time in the class of entire graphs over $${\mathbb{R}}^ n$$ without any growth assumptions near infinity for the initial surface.
Reviewer: K.Ecker

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 58J35 Heat and other parabolic equation methods for PDEs on manifolds
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