Variational principles for circle patterns and Koebe’s theorem

The subject of this paper is a special class of configurations, or patterns, of intersecting circles in constant curvature surfaces. The combinatorial aspect of such a pattern is described by a cellular decomposition of the surface. The faces of the cellular decomposition correspond to circles and the vertices correspond to points where circles intersect. (See figures 1 and 2.) In the most general case that we consider, the surface may have cone-like singularities in the centers of the circles and in the points of intersection. In particular, we treat the problem of constructing such circle patterns when the cellular decomposition is given combinatorially, the intersection angles of the circles are prescribed by a given function on the edges (and, possibly, cone angles are prescribed by functions on the faces and vertices). Using variational methods, and the ‘method of coherent angles,’ we prove existence and uniqueness results. The most fundamental one of these is theorem 3. From it we deduce Rivin’s theorem on ideal hyperbolic polyhedra [HRS92, HR93, Riv93, Riv94, Riv96], and theorem 4, which is the analogous theorem for higher genus surfaces. To our knowledge, theorem 4 is new. Our method is based on two new functionals—one for the Euclidean and one for the hyperbolic case. We show how the functionals of Colin de Verdière [CdV91], Brägger [Brä92], and Rivin [Riv94] can be derived from ours. Leibon’s functional [Lei01] seems to be related as well.