Equivariant Plateau Problems

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 $θ:π_1(Σ,P)\toπ_1(M,Q)$ 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 $i:Σ\to M$ which realises $θ$ (i.e. such that $θ=i_*$).