Global isometric embedding of a Riemannian 2-manifold with nonnegative curvature into a Euclidean 3-space
The local problem described in the title was solved by \textit{C.-S. Lin} [J. Differ. Geom. 21, 213-230 (1985; Zbl 0584.53002)]. The global problem presents new difficulties, still focussed on the set where \(K=0\). Drawing on his previous work on degenerate elliptic operators, the author proves an implicit function theorem of Nash-Moser type that is applicable to the present problem and uses it to prove the following theorem: If \(g\) is a \(C^ r\) \((r\geq 15)\) Riemannian metric on the plane with \(K\geq 0\) and \(| g_{ij}-\delta_{ij}|_ r\) sufficiently small, then on a bounded \(C^ \infty\) convex domain in the plane, \(g\) can be realized by a \(C^{(r-11)/2}\) isometric imbedding into \(R^ 3\).
scientific article; zbMATH DE number 14051