Isometric embedding in \(R^3\) of complete noncompact nonnegatively curved surfaces
This paper is a continuation of the author's work [\textit{J.-X. Hong}, Math. Z. 209, 289-306 (1992; Zbl 0771.35020) and (with \textit{C. Zuily}), ibid. 219, 323-334 (1995; Zbl 0833.53049)] on the existence of global isometric embeddings in \({\mathbb{R}}^3\) of surfaces of nonnegative curvature. Here complete noncompact surfaces are considered, and it is shown that any complete \(C^4\) metric on \({\mathbb{R}}^2\) of nonnegative curvature has a \(C^{1,1}\) isometric embedding in \({\mathbb{R}}^3\). For positive curvature metrics this problem was studied by \textit{A. V. Pogorelov} [`Extrinsic geometry of convex surfaces' (Transl. Math. Monogr. 35, AMS, Providence/RI) (1973; Zbl 0311.53067)] and others.
isometric embedding of complete noncompact surfaces
scientific article; zbMATH DE number 1135239