Remote Access Conformal Geometry and Dynamics
Green Open Access

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
DOI: https://doi.org/10.1090/S1088-4173-2010-00210-9
Published electronically: August 2, 2010
MathSciNet review: 2672225
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Mathematica 173 (1994), 155-234. MR 1301392 (95k:30046)
  • 2. M. Chrobak and D. Eppstein, Planar orientations with low out-degree and compaction of adjacency matrices, Theoretical Computer Science 86 (1991), no. 2, 243-266. MR 1122790 (93a:68114)
  • 3. P. G. Doyle and J. L. Snell, Random walks and electric networks, The Carus Mathematical Monographs, no. 22, Math. Association of America, 1984. MR 920811 (89a:94023)
  • 4. R. J. Duffin, The extremal length of a network, Journal of Mathematical Analysis and Applications 5 (1962), 200-215. MR 0143468 (26:1024)
  • 5. Z.-X. He and O. Schramm, Hyperbolic and parabolic packings, Discrete & Computational Geom. 14 (1995), 123-149. MR 1331923 (96h:52017)
  • 6. J. H. Hubbard, Teichmüller theory and application to geometry, topology, and dynamics, volume $ 1$, Matrix Editions, 2006. MR 2245223 (2008k:30055)
  • 7. B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geometry 26 (1987), 349-360. MR 906396 (90c:30007)
  • 8. W. E. Wood, Combinatorial modulus and type of refined graphs, Topology and its Applications 156 (2009), no. 17, 2747-2761, doi:10.1016/j.topol.2009.02.013. MR 2556033

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 52C26, 53A30, 05C10, 57M15

Retrieve articles in all journals with MSC (2000): 52C26, 53A30, 05C10, 57M15


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: https://doi.org/10.1090/S1088-4173-2010-00210-9
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.

American Mathematical Society