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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.

##### MSC:
 53A35 Non-Euclidean differential geometry
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