Variational principles for circle patterns and Koebe’s theorem

By Alexander I. Bobenko, Boris A. Springborn. 2002

240 citations recorded by the source corpus.

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 Verdiere's, Bragger's and Rivin's functionals can be derived from ours.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26f2d732cc1625a8bb5fbe5c89a906463b78978f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Numerical Computation of Surface Conformal Mappings (2012, methodology): “Circle pattern was proposed by Bowers and Hurdal [7], and has been proven to be a minimizer of a convex energy by Bobenko and Springborn [5].”
Discrete conformal maps and ideal hyperbolic polyhedra (2010, background): “One is due to Rivin [34] (see also the recent survey article by Futer and Guéritaud [13], which provides a wealth of material that is otherwise difficult to find), and the other is again related to it by the same sort of singular Legendre duality; see the first and third authors [6].”
COMBINATORIAL RICCI FLOWS ON SURFACES (2002, background): “In our case,it is the maximum principle and Palais-Smale condition which imply the exponential convergence. Another related work is [ 1 ].”
Face numbers of 4-Polytopes and 3-Spheres (2002, background): “(See Bobenko & Springborn [6] for a powerful treatment of this result, and for references to its involved history.)”
A variational principle for weighted Delaunay triangulations and hyperideal polyhedra (2006, methodology): “It is by no means a triviality to directly derive one type of conditions from another type [28] [6].”
Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, background): “f j dt df i dt 2 + K i df i dt2 : (12) This sort of formulation of the prescribed curvature problem in a variational framework has been studied by many people. See, for instance, [13] [36] [11] [9] [5] [41] [21]. However, no source to date has unied the theorem in the way of Theorem 23. In three di”