  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.
