The Weyl problem for unbounded convex domains in $\HH^3$

TL;DR

This paper extends the classical Weyl problem in hyperbolic space to unbounded convex domains, demonstrating that a combination of boundary conformal structure and induced metric can determine such domains under mild curvature conditions.

Contribution

It introduces a new approach to characterize unbounded convex domains in hyperbolic space using boundary conformal structure and induced metric, extending prior bounded domain results.

Findings

Existence of unbounded convex domains with prescribed boundary data

Boundary data includes conformal structure and induced metric

Mild curvature conditions suffice for realization

Abstract

Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>-1$ can be obtained. We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. Specifically, we consider the data composed of the full conformal structure on the boundary of $K$ (in the Poincar\'e model of $\HH^3$), together with the induced metric on $\partial K$. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$.