Global isometric embedding of a Riemannian 2-manifold with nonnegative curvature into a Euclidean 3-space
type: Item
instance of: scholarly article
title: Global isometric embedding of a Riemannian 2-manifold with nonnegative curvature into a Euclidean 3-space
zbMATH Open document ID: 0773.53021
author: Kazuo Amano
published in: Journal of Differential Geometry
publication date: 1992-06-25T00:00:00Z
review text: 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 \(R^ 3\).
reviewed by: Stephanie B. Alexander
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 58C15
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 14051
zbMATH Keywords: degenerate elliptic operators
zbMATH Keywords: implicit function theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214446990
OpenAlex ID: W1485227240
Wikidata QID: Q115181591
DOI: 10.4310/JDG/1214446990
description: scientific article; zbMATH DE number 14051