Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery
Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery
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双曲锥流管的刚性和双曲 Dehn 手术
DOI：
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发表时间：
1998
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通讯作者：
S. Kerckhoff
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作者：
C. Hodgson;S. Kerckhoff
WeiI [28] 和 Garland [12] 对于完整的、有限体积双曲流形的局部刚性定理指出，如果流形的维数至少为 3，则通过完整的双曲结构，这种结构不会发生非平凡的变形。如果流形是闭合的，则自动满足结构完整的条件。然而，如果流形是非紧的，则可能会因不完整的结构而产生变形。这在大于 3 的维度中不会发生（Garland-Raghunathan [13]）；但在非紧情况下，第 3 维总是存在不平凡的变形（Thurston [24]）。在本文中，我们将这种刚性和变形理论扩展到一类有限体积、可定向的 3 维双曲锥流形，即沿着结或链接具有锥状奇点的 3 流形上的双曲结构。我们的主要结果是，在所有锥角至多为 21r 的额外假设下，如果锥角固定，则此类结构是局部刚性的。我们可以将奇异结构视为奇异轨迹补集上的不完整、平滑结构，其度量完成是奇异锥结构。该开流形上结构的变形空间具有非零维数，因此在锥角保持固定的情况下也会发生变形。我们证明，可以定义结构，使度量完成仍然是一个锥流形，并且
The local rigidity theorem of WeiI [28] and Garland [12] for complete, finite volume hyperbolic manifolds states that there is no non-trivial deformation of such a structure through complete hyperbolic structures if the manifold has dimension at least 3. If the manifold is closed, the condition that the structures be complete is automatically satisfied. However, if the manifold is non-compact, there may be deformations through incomplete structures. This cannot happen in dimensions greater than 3 (Garland-Raghunathan [13]); but there are always non-trivial deformations in dimension 3 (Thurston [24]) in the non-compact case. In this paper, we extend this rigidity and deformation theory to a class of finite volume, orientable 3-dimensional hyperbolic cone-manifolds, i.e., hyperbolic structures on 3-manifolds with cone-like singularities along a knot or link. Our main result is that such structures are locally rigid if the cone angles are fixed, under the extra hypothesis that all cone angles are at most 21r. We can view the singular structure as an incomplete, smooth structure on the complement of the singular locus whose metric completion is the singular cone structure. The space of deformations of structures on this open manifold has non-zero dimension, so there will be deformations without the condition that the cone angles remain fixed. We show that, it is possible to defQrm the structure so that the metric completionJs still a cone-manifold, and that one