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MiS Preprint

Equivariant Plateau Problems

Graham Smith


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 $\theta:\pi_1(\Sigma,P)\rightarrow\pi_1(M,Q)$ 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 $i:\Sigma\rightarrow M$ which realises $\theta$ (i.e. such that $\theta=i_*$).

MSC Codes:
57M50, 30F10, 30F40, 32G15
Kleinian groups, Fuchsian groups, plateau problem, complex projective structures, immersions

Related publications

2009 Journal Open Access
Graham Smith

Equivariant Plateau problems

In: Geometric and functional analysis, 140 (2009) 1, pp. 95-135