From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a convex domain) in the 3-space form of constant curvature $\kappa$.
For $\kappa=0$, the surface is unique up to isometry of the sorrounding space $\mathbb{R}^3$. Is the same true for the elliptic and hyperbolic case? In other words, are isometric closed convex surface in these space congruent as well?