Abstract
In Theorem 1 and Proposition 2 we generalize these results of Thurston to com pact bordered triangulated surfaces. Our proof relies on Thurston's result for closed surfaces. It follows the lines of Perron's method for solving the Dirichlet Problem using subharmonic functions. In Perron's method, the harmonic solu tion is the upper envelope of a family of subharmonic functions. To actually pro duce the harmonic solution one must exhibit suitable subharmonic functions (barriers) in the Perron family. The circle packing case is analogous: we obtain the solution as the upper envelope of a Perron-type family. Nonsingular circle packings are the analogues of harmonic functions, and circle packings with neg ative curvature cone type singularities are the analogues of subharmonic functions. We use Thurston's result for closed surfaces in order to produce a negative cur vature circle packing and thereby show that the family is nontrivial. The bound ary value problem and its solution by Perron family techniques were introduced by Peter Doyle [8] and Carter and Rodin [6,7]. They treated the euclidean case for genus :51; Beardon-Stephenson [4] treated the hyperbolic simply connected case by a related method. For the results above we need a suitable notion of "circle packing on a bordered surface" when circles on the border are to have infinite hyperbolic radius. There are two obvious possibilities for such a notion: infinite border circles can be re quired to be tangent to the border or they can be required to be orthogonal to the border. That is, a hyperbolic circle of infinite radius might be defined to be ei ther a horocycle or a hyperbolic geodesic. With either choice one can define the
Published Version
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