Abstract
In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three-dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically finite hyperbolic 3-manifolds.
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References
Anderson, C. G.:Projective structures on Riemann surfaces and developing maps to ℍ3 and ℂP n, Preprint 1999.
Bers, L.:Spaces of Kleinian groups, In: Several Complex Variables I, Maryland 1970, Lecture Notes in Math. 155, Springer, 1970, pp. 9–34.
Brock, J. and Bromberg, K.:On the density of geometrically finite Kleinian groups, to appear in Acta Math.
Brock, J., Bromberg, K., Evans, R. and Souto, J.:Boundaries of deformation spaces and Ahlfors' measure conjecture, Publ. Math. IHES 98(1), 145–166.
Bromberg, K.:Hyperbolic cone-manifolds, short geodesics and Schwarzian derivatives, to appear in J. Amer. Math. Soc.
Bromberg, K.:Projective structures with degenerate holonomy and the Bers' density conjecture, Preprint 2002.
Calabi, E.: On compact Riemannian manifolds with constant curvature, I, Proc. Sympos. Pure Math. 3 (1961), 155–180.
Canary, R. D., Epstein, D. B. A. and Green, P.:Notes on notes of Thurston, In: Analytical and Geometric Aspects of Hyperbolic Space, Cambridge University Press, 1987, pp. 3–92.
Culler, M. and Shalen, P.: Varieties of group representations and splittings of 3-manifolds, Ann. of Math. 117 (1983), 109–146.
Epstein, C.:Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space, Preprint.
Gaffney, M.: A special Stokes's theorem for complete Riemannian manifolds, Ann. of Math. 60 (1954), 140–145.
Garland, H.: A rigidity theorem for discrete groups, Trans. Amer. Math. Soc. 129 (1967), 1–25.
Garland, H. and Raghunathan, M. S.: Fundamental domains for lattices in (R)-rank 1 semi-simple Lie groups, Ann. of Math. 78 (1963), 279–326.
Gunning, R.: Lectures on Vector Bundles over Riemann Surfaces, Princeton University Press, 1967.
Hejhal, D. A.: Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), 1–55.
Hodgson, C. and Kerckhoff, S.: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgey. J. Diff. Geom. 48 (1998), 1–59.
Matsushima, Y. and Murakami, S.: Vector bundle valued harmonic forms and automorphic forms on a symmetric Reimannian manifold, Ann. of Math. 78 (1963), 365–416.
McMullen, C.: Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), 283–320.
Scannell, K. and Wolf, M.: The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002), 893–927.
Thurston, W. P.: Geometry and Topology of Three-Manifolds, Princeton lecture notes, 1979.
Weil, A.: On discrete subgroups of Lie groups, Annals of Math. 72 (1960), 369–384.
Weil, A.: Remarks on the cohomology of groups, Annals of Math. 80 (1964), 149–157.
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Bromberg, K. Rigidity of Geometrically Finite Hyperbolic Cone-Manifolds. Geometriae Dedicata 105, 143–170 (2004). https://doi.org/10.1023/B:GEOM.0000024664.84428.e7
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DOI: https://doi.org/10.1023/B:GEOM.0000024664.84428.e7