  In this paper we develop methods to extend the minimal hypersurface approach
to positive scalar curvature problems to all dimensions. This includes a proof
of the positive mass theorem in all dimensions without a spin assumption. It
also includes statements about the structure of compact manifolds of positive
scalar curvature extending the work of \cite{sy1} to all dimensions. The
technical work in this paper is to construct minimal slicings and associated
weight functions in the presence of small singular sets and to show that the
singular sets do not become too large in the lower dimensional slices. It is
shown that the singular set in any slice is a closed set with Hausdorff
codimension at least three. In particular for arguments which involve slicing
down to dimension $1$ or $2$ the method is successful. The arguments can be
viewed as an extension of the minimal hypersurface regularity theory to this
setting of minimal slicings.
