The geometric data on the boundary of convex subsets of hyperbolic manifolds
Qiyu Chen, Jean-Marc Schlenker

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 be a geodesically convex subset in a convex co-compact hyperbolic manifold with incompressible boundary. We assume that each boundary component of is either a boundary component of , or a smooth, locally convex surface in . We show that 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 which meets the boundary at infinity either along a quasicircle or…
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals
