Constructing Piecewise Flat Pseudo-Manifolds with Minimal Pseudo-Foliations

Abstract

Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of non-manifold point neighborhoods is done to preserve as much of the geometric, and vector space, structure of Euclidean space as possible. This enables structures, such as foliations and calibrations, to generalize over to pseudo-manifolds. The main result is that the cone of any compact topological surfaces can be given a piecewise flat metric structure that admits a pseudo-foliation by minimal surfaces. In some situations orientation reversing holonomy is an obstruction and in others it is just the opposite, a sufficient condition. Also foliations by minimal surfaces can extend across connect sum operations.