A variational principle for weighted Delaunay triangulations and hyperideal polyhedra

Hyperideal polyhedra in hyperbolic manifolds

Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon on hyperideal polyhedra, and those of Rousset on fuchsian hyperideal polyhedra. Our results can also be stated in terms of circle configurations on $\dr M$, they provide an extension of the Koebe theorem on circle packings. The proof uses some elementary properties of the hype…

Hyperideal polyhedra in the 3-dimensional anti-de Sitter space

We study hyperideal polyhedra in the 3-dimensional anti-de Sitter space $AdS^3$, which are defined as the intersection of the projective model of $AdS^3$ with a convex polyhedron in $RP^3$ whose vertices are all outside of $AdS^3$ and whose edges all meet $AdS^3$. We show that hyperideal polyhedra in $AdS^3$ are uniquely determined by their combinatorics and dihedral angles, as well as by the induced metric on their boundary together with an additional combinatorial data, and describe the possible dihedral angles and the possible induced metrics on the boundary.

Circle patterns on singular surfaces

We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns, on Euclidean or hyperbolic surfaces with conical singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra.

球面 Delaunay 三角剖分的形变空间

The Deformation Space of Delaunay Triangulations of the Sphere

本文确定了单位二维球面上内接凸多面体空间和严格 Delaunay 测地三角剖分空间的拓扑，证实其与光滑微分同胚群具有相同同伦型，并构造平坦环面和凸多边形的反例说明相应空间不连通。

双曲端中圆模式与多面体曲面的射影刚性

Projective rigidity of circle patterns and polyhedral surfaces in hyperbolic ends

本文证明了亏格 g≥2 闭曲面上带指定组合与交角的 Delaunay 圆模式复射影结构空间是 6g-6 维流形，且遗忘映射为 Lagrangian 浸入，推广了 Bonsante-Wolf 的圆填充结果，并借助双曲端理想多面体曲面将结论拓展到指定边长及其他多面体类型。

Variational principles for circle patterns

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow cone-like singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variationa…