Compact domains with prescribed convex boundary metrics in quasi-Fuchsian manifolds

TL;DR

This paper proves the existence of convex compact domains within quasi-Fuchsian manifolds that have boundary metrics matching any given surface metric with curvature at least -1, extending geometric understanding of these manifolds.

Contribution

It establishes the existence of convex domains with prescribed boundary metrics in quasi-Fuchsian manifolds, a new result in geometric analysis.

Findings

Existence of convex domains with prescribed boundary metrics in quasi-Fuchsian manifolds.

Boundary metrics can have curvature $K \,\geq\, -1$.

Extension of Alexandrov's curvature conditions to quasi-Fuchsian settings.

Abstract

We show the existence of a convex compact domain in a quasi-Fuchsian manifold such that the induced metric on its boundary coincides with a prescribed surface metric of curvature $K\geq-1$ in the sense of A. D. Alexandrov.