Let Ft: M" ~ R "+ 1 be a hypersurface moving by its mean curvature in Euclidean space. That is, Ft=F(',t) is a one-parameter family of smooth immersions, with images Mr = Fr(M"), satisfying the evolution equationp~M", t>0.Here H(p, t)= -H(p, t).v(p, t) is the mean curvature vector of Mt at F(p, t) and v denotes a choice of unit normal for Mr. Mean curvature evolution of smooth hypersurfaces was studied previously under various global assumptions: It was shown in [8,6] that compact convex surfaces in R "+1, n>2 and embedded curves in the plane contract smoothly to a point. In [4] the authors characterized the longterm behaviour of entire graphs of controlled growth.It is the aim of this paper to study the local properties of mean curvature flow and obtain regularity estimates which are interior both in space and time.

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Abstract. Spacelike hypersurfaces of prescribed mean curvature in cosmological space times are constructed as asymptotic limits of a geometric evolution equation. In particular, an alternative, constructive proof is given for the existence of maximal and constant mean curvature slices.