Equivariant plateau problems
- 136 Downloads
Let (M, Q) be a compact, three dimensional manifold of strictly negative sectional curvature. Let (Σ, P) be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let θ : π1(Σ, P) → π1(M, Q) be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and sufficient conditions for the existence of complex projective structures with specified holonomy to manifolds of non-constant negative curvature, we obtain necessary conditions on θ for the existence of a so called θ-equivariant Plateau problem over Σ, which is equivalent to the existence of a strictly convex immersion i : Σ → M which realises θ (i.e. such that θ = i *).
KeywordsKleinian groups Fuchsian groups Plateau problem Complex projective structures Immersions
Mathematics Subject Classification (2000)57M50 30F10 30F40 32G15
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.
- 2.Ballman, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser, Boston (1985)Google Scholar
- 6.Le Roux, F.: Thèse doctorale, Grenoble (1997)Google Scholar
- 7.Smith, G.: Thèse de doctorat, Paris (2004)Google Scholar
- 8.Smith, G.: Hyperbolic Plateau Problems, Preprint, Orsay (2005)Google Scholar
- 9.Smith, G.: A Homomorphism for Braid Groups in the Sphere (in preparation)Google Scholar