Equivariant Plateau Problems

Abstract

Let $(M,Q)$ be a compact, three dimensional manifold of strictly negative sectional curvature. Let $(\Sigma ,P)$ be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let $\theta :{\pi }_1(\Sigma ,P)\to {\pi }_1(M,Q)$ 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 $\theta$ for the existence of a so called $\theta$-equivariant Plateau problem over $\Sigma$, which is equivalent to the existence of a strictly convex immersion $i:\Sigma \to M$ which realises $\theta$ (i.e. such that $\theta =i_{\ast }$).

47 pages, 2 figures, considerably shortened version containing much the same content as before