The geometric data on the boundary of convex subsets of hyperbolic manifolds

TL;DR

This paper characterizes convex subsets of hyperbolic manifolds using boundary data, extending classical theorems and providing existence results for convex domains with prescribed boundary conditions.

Contribution

It extends the hyperbolic Weyl problem and Ahlfors-Bers Theorem to convex subsets with specified boundary data, including conformal structures and fundamental forms.

Findings

Unique determination of convex subsets by boundary data

Extension of hyperbolic Weyl problem and Ahlfors-Bers Theorem

Existence results for convex domains with prescribed boundary conditions

Abstract

Let $N$ be a geodesically convex subset in a convex co-compact hyperbolic manifold $M$ with incompressible boundary. We assume that each boundary component of $N$ is either a boundary component of $\partial_\infty M$, or a smooth, locally convex surface in $M$. We show that $N$ is uniquely determined by the boundary data defined by the conformal structure on the boundary components at infinity, and by either the induced metric or the third fundamental form on the boundary components which are locally convex surfaces. We also describe the possible boundary data. This provides an extension of both the hyperbolic Weyl problem and the Ahlfors-Bers Theorem. Using this statement for quasifuchsian manifolds, we obtain existence results for similar questions for convex domains $\Omega\subset \HH^3$ which meets the boundary at infinity $\partial_{\infty}\HH^3$ either along a quasicircle or along a quasidisk. The boundary data then includes either the induced metric or the third fundamental form in $\HH^3$, but also an additional "gluing" data between different components of the boundary, either in $\HH^3$ or in $\partial_\infty\HH^3$.