  We revisit the classical problem due to Weyl, as well as its generalisations,
concerning the isometric immersions of $\mathbb{S}^2$ into simply-connected
$3$-dimensional Riemannian manifolds with non-negative Gauss curvature. A
sufficient condition is exhibited for the existence of global
$C^{1,1}$-isometric immersions. Our developments are based on the framework
\`{a} la Labourie (Immersions isom\'{e}triques elliptiques et courbes
pseudo-holomorphes, J. Diff. Geom. 30 (1989), 395--424) of studying isometric
immersions using $J$-holomorphic curves. We obtain along the way a
generalisation of a classical theorem due to Heinz and Pogorelov.
