On the boundary of convex hyperbolic manifolds.
Date(s) : 14/12/2023 iCal
11h00 - 12h00
Let S be a closed hyperbolic surface and let M = S×(0,1). Suppose 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 2π. Let L be a measured lamination on S such that every closed leaf has weight strictly less than π. 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×{0} and induces a pleated surface structure on S×{1} with bending lamination L. 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×{0}, which is viewed as one component of the ideal boundary at infinity of (M,g), and induces a pleated surface structure on S×{1} with bending lamination L. 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.
Emplacement
I2M Saint-Charles - Salle de séminaire
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