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Convex surfaces in hyperbolic space and \(\mathbb{C} P^ 1\) structures. (Surfaces convexes dans l’espace hyperbolique et \(\mathbb{C} \mathbb{P}^ 1\)- structures.) (French) Zbl 0767.53011
Let \(\mathbb{H}^ 3\) be the 3-dimensional hyperbolic space, \(S\) a compact surface of genus \(\geq 2\) and \(\tilde S\) the universal covering of \(S\). An immersion \(f: \tilde S\to \mathbb{C} P^ 1\) and a representation \(\rho: \pi_ 1(S)\to PSL(2,\mathbb{C})\) (= the group of isometries of \(\mathbb{H}^ 3\)) such that (1) \(f(\alpha\cdot s)=\rho(\alpha)f(s)\) for any \(s\in\tilde S\) and \(\alpha\in\pi_ 1(S)\) is called a \(\mathbb{C} P^ 1\)-structure on \(S\). If \(S\) is a locally convex surface of \(\mathbb{H}^ 3\), then – via the hyperbolic Gauss mapping – the existence of a \(\mathbb{C} P^ 1\)-structure on \(S\) can be shown. If \(g\) is a fixed metric on \(S\), an isometric immersion \(f: \tilde S\to\mathbb{H}^ 3\) such that there exists a representation \(\rho: \pi_ 1(S)\to PSL(2,\mathbb{C})\) satisfying (1) is called an equivariant isometric immersion of \((S,g)\) into \(\mathbb{H}^ 3\). To any equivariant isometric immersion of \((S,g)\) into \(\mathbb{H}^ 3\), a complex projective structure on \(S\) can be associated.
The author proves that the space of equivariant isometric immersions of \((S,g)\) into \(\mathbb{H}^ 3\) can be identified in two different manners with the Teichmüller space \({\mathcal T}(S)\) of \(S\). Also, two families with one parameter of parametrizations of the space of complex projective structures on \(S\) through \({\mathcal T}(S)\times {\mathcal T}(S)\) are made obvious. Some consequences are pointed out as well as the liaison with the parametrizations of H. Poincaré and W. Thurston is discussed.

53A35 Non-Euclidean differential geometry
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