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Geometriae Dedicata

, Volume 140, Issue 1, pp 95–135 | Cite as

Equivariant plateau problems

  • Graham Smith
Open Access
Original Paper

Abstract

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 *).

Keywords

Kleinian groups Fuchsian groups Plateau problem Complex projective structures Immersions 

Mathematics Subject Classification (2000)

57M50 30F10 30F40 32G15 

Notes

Open Access

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.

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Copyright information

© The Author(s) 2008

Authors and Affiliations

  • Graham Smith
    • 1
  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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