an:05174441
Zbl 1184.53049
Weiss, Hartmut
Global rigidity of 3-dimensional cone-manifolds
EN
J. Differ. Geom. 76, No. 3, 495-523 (2007).
00209573
2007
j
53C24
Summary: We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles \(\leq \pi\) (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles \(\leq \pi\) , possibly with boundary consisting of totally geodesic hyperbolic turnovers. To that end we first generalize the local rigidity result contained in [Wei] to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the techniques developed by \textit{M. Boileau, B. Leeb} and \textit{J. Porti}
[Ann. Math. (2) 162, No. 1, 195--290 (2005; Zbl 1087.57009)] to deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to Mostow-Prasad rigidit [\textit{G. D. Mostow}, Publ. Math., Inst. Hautes ??tud. Sci. 34, 53--104 (1968; Zbl 0189.09402); \textit{G. Prasad}, Invent. Math. 21, 255--286 (1973; Zbl 0264.22009)], in the hyperbolic case, resp. [\textit{G. de Rham}, in: Differ. Analysis, Bombay Colloquium 1964, 27--36 (1964; Zbl 0145.44004); cf. also [\textit{M. Rothenberg}, Proc. Symp. Pure Math., Vol. 32, Part 1, 267--311 (1978; Zbl 0426.57013)], in the spherical case. This strategy has already been implemented successfully by [Koj] in the compact hyperbolic case if the singular locus is a link using Hodgson- Kerckhoff local rigidity [\textit{C. D. Hodgson} and \textit{S. P. Kerckhoff}, J. Differ. Geom. 48, No.~1, 1--59 (1998; Zbl 0919.57009)].
Zbl 1087.57009; Zbl 0189.09402; Zbl 0264.22009; Zbl 0145.44004; Zbl 0426.57013; Zbl 0919.57009