Bounded outdegree and extremal length on discrete Riemann surfaces

Abstract:

Let $T$ be a triangulation of a Riemann surface. We show that the $1$-skeleton of $T$ may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from $T$ by attaching new edges and vertices and subdividing its faces. Such refinements provide a mechanism of convergence of the discrete triangulation to the classical surface. We will prove a bound on the distortion of the discrete extremal lengths of path families on $T$ under the refinement process. Our bound will depend only on the refinement and not on $T$. In particular, the result does not require bounded degree.