Variational principles for circle patterns and Koebe's theorem

Local rigidity of inversive distance circle packing

A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

Circle patterns with obtuse exterior intersection angles

Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the case of obtuse exterior intersection angles.

Energy of a knot: variational principles; Mm-energy

Let $E_f$ be the energy of some knot $τ$ for any $f$ from certain class of functions. The problem is to find knots with extremal values of energy. We discuss the notion of the locally perturbed knot. The knot circle minimizes some energies $E_f$ and maximizes some others. So, is there any energy such that the circle neither maximizes nor minimizes this energy? Recently it was shown (A.Abrams, J.Cantarella, J.H.G.Fu, M.Ghomu, and R.Howard) that the answer is positive. We prove that nevertheless the circle is a locally extremal knot, i.e. the circle satisfies certain variational equations. We al…

Hyper-ideal circle patterns with cone singularities

The main objective of this study is to understand how geometric hyper-ideal circle patterns can be constructed from given combinatorial angle data. We design a hybrid method consisting of a topological/deformation approach augmented with a variational principle. In this way, together with the question of characterization of hyper-ideal patterns in terms of angle data, we address their constructability via convex optimization. We presents a new proof of the main results from Jean-Marc Schlenker's work on hyper-ideal circle patterns by developing an approach that is potentially more suitable for…

A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, w…

双曲多面体曲面上反演距离圆填充的存在性

The existence of inversive distance circle packing on hyperbolic polyhedral surface

本文研究双曲多面体曲面上反演距离圆填充的存在唯一性，结合带翻边手术的离散 Ricci 流、加权 Delaunay 不等式及凸 Ricci 势变分原理完成证明。

带装饰的离散共形映射与凸多面体尖点

Decorated discrete conformal maps and convex polyhedral cusps

针对带装饰的分片欧几里得曲面，提出离散共形等价概念，证明圆不相交假设下的离散单值化定理，借助凹变分原理建立与双曲几何对象的关联并提供计算方法。

Rigidity of polyhedral surfaces, II

We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a non-triangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach on several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko-Springborn and Thurston on rigidity of polyhedral surf…