Equivariant plateau problems
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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
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