Combinatorial Minimal Surfaces in Pseudomanifolds and Other Complexes

Abstract

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds and their generalizations. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called \emph{thin position}. Thin position is defined using orderings of the $n$-dimensional simplices of an $n$-dimensional pseudomanifold. In addition to defining and finding combinatorial minimal surfaces, from thin orderings, we derive invariants of even-dimensional closed simplicial pseudomanifolds called \emph{width} and \emph{trunk}. We study additivity properties of these invariants under connected sum and prove theorems analogous to those in knot theory and 3-manifold theory.