I am interested in prescribing the boundary for $\mathbb{R}^3$-embeddings of simply connected closed subsets of the hyperbolic plane. In particular, can a hyperbolic disk isometrically embed into $\mathbb{R}^3$, and have a circle for its boundary?
This is doable for spherical disks, so maybe it could be possible for hyperbolic disks.
If it is not possible then perhaps a weaker statement holds. Can a circle in $\mathbb{R}^3$ be the boundary of a simply connected surface of constant negative Gaussian curvature?