The deformation theory of hyperbolic cone–3–manifolds with cone-angles less than 2π

Weiß, Hartmut¹

¹ LMU München, Mathematisches Institut, Theresienstr. 39, D-80333 München, Germany

Geometry & topology, Tome 17 (2013) no. 1, pp. 329-367.

Résumé

We develop the deformation theory of hyperbolic cone–3–manifolds with cone-angles less than $2\pi$, that is, contained in the interval $\left(0,2\pi \right)$. 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.

Détail

DOI : 10.2140/gt.2013.17.329

Classification : 53C25, 57M50
Keywords: cone-manifolds, geometric structures on low-dimensional manifolds, hyperbolic geometry

Bibliographie

[1] M Boileau, B Leeb, J Porti, Geometrization of 3–dimensional orbifolds, Ann. of Math. 162 (2005) 195

[2] J Brüning, M Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992) 88

[3] J Brüning, R Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988) 659

[4] A Casson, An example of weak non-rigidity for cone manifolds with vertices, Talk at the third MSJ regional workshop, Tokyo (1998)

[5] 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. (1980) 91

[6] J Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983) 575

[7] D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, 5, Mathematical Society of Japan (2000)

[8] J B Gil, G A Mendoza, Adjoints of elliptic cone operators, Amer. J. Math. 125 (2003) 357

[9] C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1

[10] M Kapovich, Hyperbolic manifolds and discrete groups, 183, Birkhäuser (2001)

[11] M Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, 136, B. G. Teubner Verlagsgesellschaft mbH (1997) 190

[12] F Luo, G Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992) 1119

[13] Y Matsushima, S Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. 78 (1963) 365

[14] R Mazzeo, Elliptic theory of differential edge operators I, Comm. Partial Differential Equations 16 (1991) 1615

[15] R Mazzeo, G Montcouquiol, Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra, J. Differential Geom. 87 (2011) 525

[16] R Mazzeo, H Weiss, Teichmüller theory for conic surfaces, in preparation

[17] R B Melrose, The Atiyah–Patodi–Singer index theorem, 4, A K Peters Ltd. (1993)

[18] R B Melrose, G Mendoza, Elliptic operators of totally characteristic type, MSRI preprint (1983)

[19] G Montcouquiol, Deformations of hyperbolic convex polyhedra and cone-3-manifolds, Geom. Dedicata (to appear)

[20] 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

[21] J Porti, H Weiss, Deforming Euclidean cone 3–manifolds, Geom. Topol. 11 (2007) 1507

[22] B W Schulze, Pseudo-differential operators on manifolds with singularities, 24, North-Holland Publishing Co. (1991)

[23] M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793

[24] H Weiss, Local rigidity of 3–dimensional cone-manifolds, J. Differential Geom. 71 (2005) 437