Projective rigidity of circle patterns and polyhedral surfaces in hyperbolic ends

Let $S$ be a closed, orientable surface of genus $g\geq 2$. We consider Delaunay circle patterns on $S$ equipped with a complex projective structure. We prove that the space of complex projective structures on $S$ equipped with a Delaunay circle pattern of prescribed combinatorics and intersection angles is a manifold of dimension $6g-6$, and that the forgetful map to the space $\cC_S$ of $\CP^1$-structures on $S$ is a Lagrangian immersion. This extends a recent result of Bonsante and Wolf for circle packings. This statement, and its proof, are more conveniently stated in terms of ideal polyhedral surfaces (surfaces with vertices at infinity) in hyperbolic ends, with the angles between the circles corresponding to the dihedral angles. Seen from this angle, we extend the statement to ideal polyhedral surfaces with prescribed edge lengths (or induced metrics), and to other types of polyhedral surfaces, either compact or hyperideal.