Isometric embedding in \(R^3\) of complete noncompact nonnegatively curved surfaces
type: Item
instance of: scholarly article
title: Isometric embedding in \(R^3\) of complete noncompact nonnegatively curved surfaces
zbMATH Open document ID: 0894.53050
DOI: 10.1007/BF02677853
published in: Manuscripta Mathematica
publication date: 1998-08-30T00:00:00Z
full work available at URL: https://eudml.org/doc/156338
review text: 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.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1135239
zbMATH Keywords: isometric embedding of complete noncompact surfaces
zbMATH Keywords: nonnegative curvature metrics
author: Jiaxing Hong
reviewed by: John Urbas
MaRDI profile type: Publication
cites work: Q5797644
cites work: Q4067705
cites work: On the regularity of the solution of then-dimensional Minkowski problem
cites work: Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length
cites work: The Weyl problem with nonnegative Gauss curvature
cites work: Integrals of subharmonic functions on manifolds of nonnegative curvature
cites work: \(C^\infty\) convex functions and manifolds of positive curvature
cites work: Q5599507
cites work: On elliptic Monge-Ampère equations and Weyl's embedding problem
cites work: Neue a-priori-Abschätzungen für den Ortsvektor einer Fläche positiver Gaußscher Krümmung durch ihr Linienelement
cites work: Dirichlet problems for general Monge-Ampère equations
cites work: Isometric embedding of the 2-sphere with non negative curvature in \(\mathbb{R}^ 3\)
cites work: Isometric embeddings of surfaces with nonnegative curvature in \(\mathbb{R}^ 3\)
cites work: The local isometric embedding inR3 of two-dimensional riemannian manifolds with gaussian curvature changing sign cleanly
cites work: The Weyl and Minkowski problems in differential geometry in the large
cites work: On nonlinear elliptic partial differential equations and hölder continuity
cites work: Q4069454
OpenAlex ID: W1496081729
description: scientific article; zbMATH DE number 1135239