Geometry & Topology

The deformation theory of hyperbolic cone–$3$–manifolds with cone-angles less than $2\pi$

Hartmut Weiß

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Abstract

We develop the deformation theory of hyperbolic cone–3–manifolds with cone-angles less than 2π, that is, contained in the interval (0,2π). In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a question of A Casson.

Article information

Source
Geom. Topol., Volume 17, Number 1 (2013), 329-367.

Dates
Received: 12 April 2012
Accepted: 9 September 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732526

Digital Object Identifier
doi:10.2140/gt.2013.17.329

Mathematical Reviews number (MathSciNet)
MR3035330

Zentralblatt MATH identifier
1262.53032

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
cone-manifolds geometric structures on low-dimensional manifolds hyperbolic geometry

Citation

Weiß, Hartmut. The deformation theory of hyperbolic cone–$3$–manifolds with cone-angles less than $2\pi$. Geom. Topol. 17 (2013), no. 1, 329--367. doi:10.2140/gt.2013.17.329. https://projecteuclid.org/euclid.gt/1513732526


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References

  • M Boileau, B Leeb, J Porti, Geometrization of 3–dimensional orbifolds, Ann. of Math. 162 (2005) 195–290
  • J Brüning, M Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992) 88–132
  • J Brüning, R Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988) 659–714
  • A Casson, An example of weak non-rigidity for cone manifolds with vertices, Talk at the third MSJ regional workshop, Tokyo (1998)
  • J Cheeger, On the Hodge theory of Riemannian pseudomanifolds, from: “Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979)”, Proc. Sympos. Pure Math. XXXVI, Amer. Math. Soc., Providence, R.I. (1980) 91–146
  • J Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983) 575–657
  • D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan, Tokyo (2000) With a postface by Sadayoshi Kojima
  • J B Gil, G A Mendoza, Adjoints of elliptic cone operators, Amer. J. Math. 125 (2003) 357–408
  • C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1–59
  • M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics 183, Birkhäuser, Boston, MA (2001)
  • M Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Mathematik 136, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart (1997)
  • F Luo, G Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992) 1119–1129
  • Y Matsushima, S Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. 78 (1963) 365–416
  • R Mazzeo, Elliptic theory of differential edge operators I, Comm. Partial Differential Equations 16 (1991) 1615–1664
  • R Mazzeo, G Montcouquiol, Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra, J. Differential Geom. 87 (2011) 525–576
  • R Mazzeo, H Weiss, Teichmüller theory for conic surfaces, in preparation
  • R B Melrose, The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics 4, A K Peters Ltd., Wellesley, MA (1993)
  • R B Melrose, G Mendoza, Elliptic operators of totally characteristic type, MSRI preprint (1983)
  • G Montcouquiol, Deformations of hyperbolic convex polyhedra and cone-3-manifolds, Geom. Dedicata (to appear)
  • G Montcouquiol, H Weiss, Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone-3–manifolds, Geom. Topol. 17 (2013) 369
  • J Porti, H Weiss, Deforming Euclidean cone 3–manifolds, Geom. Topol. 11 (2007) 1507–1538
  • B-W Schulze, Pseudo-differential operators on manifolds with singularities, Studies in Mathematics and its Applications 24, North-Holland Publishing Co., Amsterdam (1991)
  • M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821
  • H Weiss, Local rigidity of 3–dimensional cone-manifolds, J. Differential Geom. 71 (2005) 437–506