The induced metric and bending lamination on the boundary of convex hyperbolic 3‐manifolds

Abstract Let be an oriented closed surface of genus at least two, and let . Suppose that is a Riemannian metric on with curvature strictly greater than , is a Riemannian metric on with curvature strictly less than 1, and every contractible closed geodesic with respect to has length strictly greater than . Let be a measured lamination on such that every closed leaf has weight strictly less than . Then, we prove the existence of a convex hyperbolic metric on the interior of that induces the Riemannian metric (respectively, ) as the first (respectively, third) fundamental form on and induces a pleated surface structure on with bending lamination . This statement remains valid even in limiting cases where the curvature of is constant and equal to . In addition, when considering a conformal class on , we show that there exists a convex hyperbolic metric on the interior of that induces on , which is viewed as one component of the ideal boundary at infinity of , and induces a pleated surface structure on with bending lamination . Our proof differs from previous work by Lecuire for these two last cases. Moreover, when we consider a lamination which is small enough, in a sense that we will define, and a hyperbolic metric, we show that the metric on the interior of M that realizes these data is unique.