zbMATH — the first resource for mathematics

Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone-3-manifolds. (English) Zbl 1267.57021
Hyperbolic cone-3-manifolds are hyperbolic manifolds with singular locus an embedded graph and a conical metric around the singularity. They can be constructed by gluing hyperbolic simplices by isometries along faces, so that the singular locus is contained in the 1-skeleton.
Local rigidity of three-dimensional cone-manifolds with (fixed) cone angles less than \(2\pi\) has been established by several authors, provided that the topology of the singular graph is preserved: namely the space of deformations is locally parameterized by the cone angles. Among others we cite C. D. Hodgson and S. P. Kerckhoff [J. Differ. Geom. 48, No. 1, 1–59 (1998; Zbl 0919.57009)], R. Mazzeo and G. Montcouquiol [ibid. 87, No. 3, 525–576 (2011; Zbl 1234.53014)], and H. Weiss [ibid. 71, No. 3, 437–506 (2005; Zbl 1098.53038)].
When the cone angles are less than \(\pi\), the topology of the singular locus does not change under deformation. This paper however shows that when the angles are larger than \(\pi\), then there exist deformations such that the vertices of the singular locus of valency \(\geq 4\) can split along new singular edges. To describe such splittings one has to look at the link of a singular vertex (which is a sphere with cone points, one for each singular edge adjacent to the vertex). A splitting of the vertex is described in terms of curves in the link, so that the corresponding edges are split by this curve. The authors give sufficient conditions in terms of the geometry of this curve so that the splitting exists. Moreover they give precise parameters for the deformation space with this given topology, namely the cone angles of the old singular edges and the length of the new edges.
The result uses a cohomology theorem of Weiss, and the paper mainly deals with representation spaces. In particular, it relies on the results of W. M. Goldman [in: Dani, S. G. (ed.) et al., Algebraic groups and arithmetic. Proceedings of the international conference, Mumbai, India, December 17–22, 2001. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research. 375–407 (2004; Zbl 1089.53060)] on the complex symplectic structure of the representations of a surface in \(SL(2,\mathbb C)\) as well as the geometric interpretation of certain infinitesimal deformations.

57M50 General geometric structures on low-dimensional manifolds
58D27 Moduli problems for differential geometric structures
53C35 Differential geometry of symmetric spaces
Full Text: DOI arXiv
[1] M F Atiyah, R Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523 · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017
[2] M Boileau, B Leeb, J Porti, Geometrization of \(3\)-dimensional orbifolds, Ann. of Math. 162 (2005) 195 · Zbl 1087.57009 · doi:10.4007/annals.2005.162.195 · euclid:annm/1134073584
[3] D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan (2000) · Zbl 0955.57014
[4] W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200 · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9
[5] W M Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986) 263 · Zbl 0619.58021 · doi:10.1007/BF01389091 · eudml:143369
[6] W M Goldman, Geometric structures on manifolds and varieties of representations (editors W M Goldman, A R Magid), Contemp. Math. 74, Amer. Math. Soc. (1988) 169 · Zbl 0659.57004 · doi:10.1090/conm/074/957518
[7] W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557 · Zbl 0655.57019 · doi:10.1007/BF01410200 · eudml:143609
[8] W M Goldman, The complex-symplectic geometry of \(\mathrm{SL}(2,\mathbb C)\)-characters over surfaces (editors S G Dani, G Prasad), Tata Inst. Fund. Res. (2004) 375
[9] W J Harvey, Boundary structure of the modular group (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245 · Zbl 0461.30036
[10] C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1 · Zbl 0919.57009 · euclid:jdg/1214460606
[11] W P A Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics 1, Walter de Gruyter & Co. (1995) · Zbl 0911.53022 · doi:10.1515/9783110905120
[12] K Krasnov, J M Schlenker, Minimal surfaces and particles in \(3\)-manifolds, Geom. Dedicata 126 (2007) 187 · Zbl 1126.53037 · doi:10.1007/s10711-007-9132-1 · arxiv:math/0511441
[13] F Luo, G Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992) 1119 · Zbl 0806.53012 · doi:10.2307/2159498
[14] H A Masur, Y N Minsky, Geometry of the complex of curves, I, Hyperbolicity, Invent. Math. 138 (1999) 103 · Zbl 0941.32012 · doi:10.1007/s002220050343 · arxiv:math/9804098
[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 · Zbl 1234.53014 · euclid:jdg/1312998235
[16] R Mazzeo, H Weiss, Teichmüller theory for conic surfaces, in preparation · Zbl 1444.32015
[17] G Montcouquiol, Deformations of hyperbolic convex polyhedra and cone-\(3\)-manifolds, to appear in Geom. Dedicata (2013) · Zbl 1279.52015 · doi:10.1007/s10711-012-9790-5
[18] G D Mostow, Quasi-conformal mappings in \(n\)-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. (1968) 53 · Zbl 0189.09402 · doi:10.1007/BF02684590 · numdam:PMIHES_1968__34__53_0 · eudml:103882
[19] J Porti, Regenerating hyperbolic cone structures from Nil, Geom. Topol. 6 (2002) 815 · Zbl 1032.57015 · doi:10.2140/gt.2002.6.815 · emis:journals/UW/gt/GTVol6/paper24.abs.html · eudml:123317 · arxiv:math/0212298
[20] J Porti, H Weiss, Deforming Euclidean cone \(3\)-manifolds, Geom. Topol. 11 (2007) 1507 · Zbl 1159.57007 · doi:10.2140/gt.2007.11.1507 · arxiv:math/0510432
[21] J M Schlenker, Dihedral angles of convex polyhedra, Discrete Comput. Geom. 23 (2000) 409 · Zbl 0951.52006 · doi:10.1007/PL00009509
[22] J J Stoker, Geometrical problems concerning polyhedra in the large, Comm. Pure Appl. Math. 21 (1968) 119 · Zbl 0159.24301 · doi:10.1002/cpa.3160210203
[23] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) · msri.org
[24] W P Thurston, Earthquakes in two-dimensional hyperbolic geometry (editor D B A Epstein), London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press (1986) 91 · Zbl 0628.57009
[25] W P Thurston, Shapes of polyhedra and triangulations of the sphere (editors I Rivin, C Rourke, C Series), Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 511 · Zbl 0931.57010 · doi:10.2140/gtm.1998.1.511 · emis:journals/UW/gt/GTMon1/paper25.abs.html
[26] V A Toponogov, Evaluation of the length of a closed geodesic on a convex surface, Dokl. Akad. Nauk SSSR 124 (1959) 282 · Zbl 0092.14603
[27] M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793 · Zbl 0724.53023 · doi:10.2307/2001742
[28] A Weil, On discrete subgroups of Lie groups, Ann. of Math. 72 (1960) 369 · Zbl 0131.26602 · doi:10.2307/1970140
[29] H Weiss, Local rigidity of \(3\)-dimensional cone-manifolds, J. Differential Geom. 71 (2005) 437 · Zbl 1098.53038 · euclid:jdg/1143571990
[30] H Weiss, Global rigidity of \(3\)-dimensional cone-manifolds, J. Differential Geom. 76 (2007) 495 · Zbl 1184.53049 · euclid:jdg/1180135696
[31] H Weiss, The deformation theory of hyperbolic cone-\(3\)-manifolds with cone-angles less than \(2\pi\), Geom. Topol. 17 (2013) 329 · Zbl 1262.53032 · doi:10.2140/gt.2013.17.329
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.