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The upper Perron method for labelled complexes with applications to circle packings

Published online by Cambridge University Press:  24 October 2008

Philip L. Bowers
Philip L. Bowers
Affiliation:
Department of Mathematics, The Florida State University, Tallahassee, FL 32306-3027, U.S.A.
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Extract

The construction of geometric surfaces via labelled complexes was introduced by Thurston[16, chapter 13], and subsequent applications and developments have appeared in [1, 3, 4, 5, 14, 15]. The basic idea of using labelled complexes to produce geometric structures is that the vertices of a simplicial triangulation of a surface can be labelled with positive real numbers that collectively determine a metric of constant curvature ±1 or 0, with possible singularities at vertices, by using the label values to identify 2-simplices of the triangulation with geometric triangles. Beardon and Stephenson[1] developed a particularly simple method for producing non-singular surfaces via labelled complexes that is modelled after the classical Perron method for producing harmonic functions, and they applied their method in [2] to construct a fairly comprehensive theory of circle packings in general Riemann surfaces. This Perron method was developed more fully by Stephenson and the author in [3, 4] and applied to the study of circle packing points in moduli space. At about the same time and independently of Beardon, Stephenson, and Bowers, Carter and Rodin [5] and Doyle [8] developed the method for flat surfaces and Minda and Rodin [14] developed the method for finite type surfaces. Minda and Rodin [14] applied their development to give partial solutions to the labelled complex version of the classical Schwarz-Picard problem that concerns the construction of singular hyperbolic metrics on surfaces with prescribed singularities. In this paper, we modify the aforementioned approaches and examine the upper Perron method for producing non-singular geometric surfaces. This upper method has several advantages over the Perron method as developed previously and provides a complete solution to the labelled complex version of the Schwarz-Picard problem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 2 , September 1993 , pp. 321 - 345
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

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