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

Abstract

Let $S$ be an oriented closed surface of genus at least two, and let $M = S \times (0,1)$. Suppose that $h$ is a Riemannian metric on $S$ with curvature strictly greater than $-1$, $h^{*}$ is a Riemannian metric on $S$ with curvature strictly less than $1$, and every contractible closed geodesic with respect to $h^{*}$ has length strictly greater than $2\pi$. Let $\mu$ be a measured lamination on $S$ such that every closed leaf has weight strictly less than $\pi$. Then, we prove the existence of a convex hyperbolic metric $g$ on the interior of $M$ that induces the Riemannian metric $h$ (respectively $h^{*}$) as the first (respectively third) fundamental form on $S \times \left\{ 0\right\}$ and induces a pleated surface structure on $S \times \left\{ 1\right\}$ with bending lamination $\mu$. This statement remains valid even in limiting cases where the curvature of $h$ is constant and equal to $-1$. Additionally, when considering a conformal class $c$ on $S$, we show that there exists a convex hyperbolic metric $g$ on the interior of $M$ that induces $c$ on $S \times \left\{ 0\right\}$, which is viewed as one component of the ideal boundary at infinity of $(M,g)$, and induces a pleated surface structure on $S \times \left\{ 1\right\}$ with bending lamination $\mu$. 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.