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 (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with is constructed – thus, the spacetime splits orthogonally as 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).