分段常曲率曲面的对偶结构与离散共形变分
Duality structures and discrete conformal variations of piecewise constant curvature surfaces
摘要：
分段常曲率流形是通过指定边的长度并规定对于所选背景几何（欧几里得、双曲或球面），每个单形具有等距嵌入到具有所选边长度的背景几何中来分配几何的三角化流形。额外的结构是通过给三角剖分的庞加莱对偶一个几何结构或通过分配一个离散度量来定义的，这是一种将长度分配给定向边的方法。这个概念导致了离散共形结构的概念，推广了基于圆填充的离散共形结构及其由Thurston等人研究的推广。定义并分析了分片常曲率2-流形的共形变分，特别注意角的变化。我们给公式的角度在每个背景的几何形状，从而产生公式的导数的曲率。我们的公式使我们能够确定特定的曲率泛函与共形变化。最后，我们提供了一个完整的分类的离散共形结构在每个背景的几何形状。
英文摘要：
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Additional structure is defined either by giving a geometric structure to the Poincare dual of the triangulation or by assigning a discrete metric, a way of assigning length to oriented edges. This notion leads to a notion of discrete conformal structure, generalizing the discrete conformal structures based on circle packings and their generalizations studied by Thurston and others. We define and analyze conformal variations of piecewise constant curvature 2-manifolds, giving particular attention to the variation of angles. We give formulas for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures. Our formulas allow us to identify particular curvature functionals associated with conformal variations. Finally, we provide a complete classification of discrete conformal structures in each of the background geometries.
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