We revisit the classical problem by 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 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.

中文翻译：

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等距浸没的Weyl问题

我们回顾了Weyl的经典问题及其推广，涉及到等距浸入 ${\mathbb{小号}}^2$进入具有非负高斯曲率的简单连接的3维黎曼流形。展示了存在全球的充分条件$C^{\mathrm{1个}，\mathrm{1个}}$等距浸入。我们的开发基于àla Labourie框架（J. Differential Geom。30（1989）395–424），该框架可通过以下方式分析等距浸入$Ĵ$全纯曲线。一路上，我们得到了归因于亨氏和波哥列洛夫的一个著名定理的推广。