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.

9 pages, 2 figures

References in corpus (1)

• Combinatorial modulus and type of graphs

Cited by in corpus (2)

• Discrete Extremal Length and Cube Tilings in Finite Dimensions
• Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube