Polyedrische 2-Mannigfaltigkeiten mit wenigen nicht-konvexen Ecken

Let P denote a polyhedral 2-manifold in ℝ³, i.e. a 2-dimensional cell-complex in ℝ³ whose underlying point-set is a closed connected 2-manifold. A vertex v of P is called convex if at least one of the two components into which P divides a sufficiently small ball centered at v is convex. It is shown that every polyhedral 2-manifold in ℝ³ of genus g>-1 contains at least five non-convex vertices and that for every positive integer g this bound is attained, i.e. there exists a polyhedral 2-manifold in ℝ³ of genus g with precisely five non-convex vertices.