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 *).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Smith, G. Equivariant plateau problems. Geom Dedicata 140, 95–135 (2009). https://doi.org/10.1007/s10711-008-9310-9
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DOI: https://doi.org/10.1007/s10711-008-9310-9