Faculty Publications

Bounded Outdegree and Extremal Length on Discrete Riemann Surfaces

Document Type

Article

Journal/Book/Conference Title

Conformal Geometry and Dynamics

Volume

14

First Page

194

Last Page

201

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.

Department

Department of Mathematics

Original Publication Date

8-2-2010

DOI of published version

10.1090/S1088-4173-2010-00210-9

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