The Weyl problem of isometric immersions revisited

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.