The shape of hyperbolic Dehn surgery space
The shape of hyperbolic Dehn surgery space
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DOI：
10.2140/gt.2008.12.1033
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发表时间：
2007-09
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影响因子：
--
通讯作者：
C. Hodgson;S. Kerckhoff
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文献类型：
--
作者：
C. Hodgson;S. Kerckhoff
本文发展了一个关于具有“管状边界”的紧致双曲三维流形的无穷小调和形变的新理论。特别地，这适用于半径至少为$R_0 =\arctanh（1/\sqrt {3}）\approximat0.65848 $的管的补数，去除了之前对锥角的限制。然后，我们应用这一点，以获得一个新的定量版本的瑟斯顿的双曲德恩手术定理，表明所有广义德恩手术系数以外的光盘的“统一”的大小产生双曲结构。在这里，手术系数的大小是使用欧氏度量在完全双曲度量中的尖点的半球形横截面上测量的，重新缩放为具有面积1。我们还获得了良好的估计在双曲Dehn填充过程中的几何变化（例如体积和核心测地线长度）。这个新的调和变形理论也被布朗伯格和他的同事们用于证明克莱因群的Bers密度猜想。
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a surgery coefficient is measured using the Euclidean metric on a horospherical cross section to a cusp in the complete hyperbolic metric, rescaled to have area 1. We also obtain good estimates on the change in geometry (e.g. volumes and core geodesic lengths) during hyperbolic Dehn filling. This new harmonic deformation theory has also been used by Bromberg and his coworkers in their proofs of the Bers Density Conjecture for Kleinian groups.