Bounded Outdegree and Extremal Length on Discrete Riemann Surfaces

TL;DR

This paper demonstrates how to orient the 1-skeleton of a triangulated Riemann surface to achieve a uniform outdegree bound, enabling convergence analysis of discrete extremal metrics to classical surface metrics.

Contribution

It introduces a method to orient triangulation graphs with bounded outdegree and constructs extremal metrics that ensure convergence of discrete to classical surfaces without degree restrictions.

Findings

Bound on outdegree of triangulation vertices achieved

Constructed extremal metrics facilitate convergence analysis

Bound on extremal length distortion independent of initial degree

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.