圆模式的变分原理
Variational principles for circle patterns
摘要：
具有恒定曲率的表面的Delaunay胞腔剖分产生圆图案，其由外切到小平面的圆组成。我们对待的问题是否存在一个Delaunay胞腔剖分为一个给定的（拓扑）胞腔剖分和给定的交叉角的圆圈，它是否是唯一的，以及它如何可能被构造。更一般地说，我们允许在圆心和交点上有锥状奇点。利用变分原理证明了圆模式问题解的存在唯一性定理。泛函（一个用于欧几里德，一个用于双曲情形）是圆半径的凸函数。球面情形的类似泛函不是凸的，因此这种情形用到平面的球极平面投影来处理。从球面上圆图案的存在唯一性出发，得到了Steinitz关于抽象多面体几何可实现性定理的一个加强形式。从我们的变分原理出发，导出了Colin de Verdière，Brägger和Rivin关于圆填充和圆模式的变分原理.在Brägger和Rivin泛函的情况下。Leibon的双曲圆模式的功能不能直接从我们的泛函。但是，我们构造另一个功能，从这两个Leibon的和我们的泛函可以得出。我们提出了Java软件来计算和可视化圆图案。
英文摘要：
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 variational principle. The functionals (one for the euclidean, one for the hyperbolic case) are convex functions of the radii of the circles. The analogous functional for the spherical case is not convex, hence this case is treated by stereographic projection to the plane. From the existence and uniqueness of circle patterns in the sphere, we derive a strengthened version of Steinitz' theorem on the geometric realizability of abstract polyhedra. We derive the variational principles of Colin de Verdière, Brägger, and Rivin for circle packings and circle patterns from our variational principles. In the case of Brägger's and Rivin's functionals. Leibon's functional for hyperbolic circle patterns cannot be derived directly from our functionals. But we construct yet another functional from which both Leibon's and our functionals can be derived. We present Java software to compute and visualize circle patterns.
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