A note on circle patterns on surfaces (Q2642388)

Let \(F\) be the set of faces and \(E\) the set of edges of a cellular decomposed surface \(\Sigma\). Prescribing a radius \(\Phi(f)\) for each cell \(f\), and an angle \(D(e)\) for each edge e at which the two circumcircles of the cells adjacent to \(e\) shall intersect, there exists a unique circle pattern, i.e. a hyperbolic on an Euclidean structure on \(\Sigma\) if and only if for all non-empty subsets \(X\subseteq F\) \(\sum_{f\in X}{1\over 2}\Phi(f)< \sum_{e\in E} D(e)\), resp. if and only if for all non-empty subsets \(X\subseteq F\) \(\sum_{f\in F}{1\over 2}\Phi(f)\leq \sum_{e\in E}D(e)\) with equality just if \(X= F\). This is the theorem of Bubenko and Springborn, of which the author offers here two new proofs, based on the continuity method and on a variational principle.