Complex earthquakes and deformations of the unit disk
Complex earthquakes and deformations of the unit disk
复制标题
复杂地震和单位圆盘变形
DOI：
10.4310/jdg/1146680514
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发表时间：
2006
影响因子：
2.5
通讯作者：
V. Marković
中科院分区：
文献类型：
--
作者：
D. Epstein;A. Marden;V. Marković
本文定义了三维双曲空间中某些几何对象的形变。这样一个物体开始时是一个双曲平面，有一个测量过的几何分层。最初的双曲平面嵌入作为一个标准的双曲子空间。给定一个复数t，我们通过沿着纹理进行地震（由t的真实的部参数化），然后沿着图像纹理（由t的复部参数化）进行沿着弯曲，在双曲三维空间中获得相应的物体。在文献中，通常假设存在一个保持结构的拟fuchsian群，但本文更一般，没有做这样的假设。我们的变形是全纯的，如λ-引理，这是本文结果的基础。我们的变形是用来产生一个新的，更自然的苏利文定理的证明：即，在标准的拓扑假设，边界的凸船体在双曲3-空间的补的一个开放的子集U的2-球是准共形等价于U，而且，此外，常数的准共形是一个普遍常数。本文给出了Sullivan定理的一个精确表述。我们还推广了麦克马伦圆盘定理，描述了某些参数化空间的二维双曲结构的参数空间的某些方面。
We define deformations of certain geometric objects in hyperbolic 3-space. Such an object starts life as a hyperbolic plane with a measured geometric lamination. Initially the hyperbolic plane is embedded as a standard hyperbolic subspace. Given a complex number t, we obtain a corresponding object in hyperbolic 3-space by earthquaking along the lamination, parametrized by the real part of t, and then bending along the image lamination, parametrized by the complex part of t. In the literature, it is usually assumed that there is a quasifuchsian group that preserves the structure, but this paper is more general and makes no such assumption. Our deformation is holomorphic, as in the λ-lemma, which is a result that underlies the results in this paper. Our deformation is used to produce a new, more natural proof of Sullivan's theorem: that, under standard topological hypotheses, the boundary of the convex hull in hyperbolic 3-space of the complement of an open subset U of the 2-sphere is quasi-conformally equivalent to U, and that, furthermore, the constant of quasiconformality is a universal constant. Our paper presents a precise statement of Sullivan's Theorem. We also generalize much of McMullen's Disk Theorem, describing certain aspects of the parameter space for certain parametrized spaces of 2-dimensional hyperbolic structures.