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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

Bounded outdegree and extremal length on discrete Riemann surfaces


Author: William E. Wood
Journal: Conform. Geom. Dyn. 14 (2010), 194-201
MSC (2000): Primary 52C26; Secondary 53A30, 05C10, 57M15
Published electronically: August 2, 2010
MathSciNet review: 2672225
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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.


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Additional Information

William E. Wood
Affiliation: Department of Mathematics and Computer Science, 1600 Washington Avenue, Hendrix College, Conway, Arkansas 72032
Email: wood@hendrix.edu

DOI: http://dx.doi.org/10.1090/S1088-4173-2010-00210-9
PII: S 1088-4173(2010)00210-9
Keywords: Discrete conformal geometry, extremal length
Received by editor(s): September 1, 2009
Published electronically: August 2, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.