Variational principles for circle patterns and Koebe’s theorem

Bobenko, Alexander; Springborn, Boris

doi:10.1090/S0002-9947-03-03239-2

Variational principles for circle patterns and Koebe’s theorem
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by Alexander I. Bobenko and Boris A. Springborn PDF

Trans. Amer. Math. Soc. 356 (2004), 659-689 Request permission

Abstract:

We prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces. Our method is based on two new functionals—one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdière’s, Brägger’s and Rivin’s functionals can be derived from ours.

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