Combinatorial minimal surfaces in pseudomanifolds

TL;DR

This paper introduces combinatorial analogues of minimal surfaces within weighted pseudomanifolds, establishing their existence, and develops invariants like width and trunk using thin position, with applications to topological properties.

Contribution

It defines combinatorial minimal surfaces in pseudomanifolds, proves their existence, and introduces new invariants derived from thin position techniques.

Findings

Existence of combinatorial minimal surfaces under mild conditions.

Introduction of width and trunk invariants for pseudomanifolds.

Additivity properties of invariants under connected sum.

Abstract

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