Further results on the smoothability of cauchy hypersurfaces and Cauchy time functions

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension ( in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S=T-1( 0) is constructed - thus, the spacetime splits orthogonally as R x S in a canonical way. Even more, accurate versions of this last result are obtained if the Cauchy hypersurface S were non-spacelike ( including non-smooth, or achronal but non-acausal).