The Weyl problem of isometric immersions revisited

Abstract

We revisit the classical problem by Weyl, as well as its generalisations, concerning the isometric immersions of S2 into simply‐connected 3‐dimensional Riemannian manifolds with non‐negative Gauss curvature. A sufficient condition is exhibited for the existence of global C1,1‐isometric immersions. Our developments are based on the framework à la Labourie (J. Differential Geom. 30 (1989) 395–424) of analysing isometric immersions via J‐holomorphic curves. We obtain along the way a generalisation of a well‐known theorem due to Heinz and Pogorelov.