Isometric embedding of the 2-sphere with non negative curvature in \(\mathbb{R}^ 3\)
type: Item
instance of: scholarly article
title: Isometric embedding of the 2-sphere with non negative curvature in \(\mathbb{R}^ 3\)
zbMATH Open document ID: 0833.53049
DOI: 10.1007/BF02572368
published in: Mathematische Zeitschrift
publication date: 1995-10-22T00:00:00Z
full work available at URL: https://eudml.org/doc/174776
review text: In 1916 H. Weyl asked whether \(S^2\) equipped with a Riemannian metric \(g\) of positive Gauss curvature \(K(g)\) can be isometrically embedded in \(\mathbb{R}^3\) equipped with the standard metric. Affirmative answers for metrics of various degrees of smoothness were provided by Lewy, Nirenberg, Heinz, Aleksandrov and Pogorelov. Recently \textit{J. A. Iaia} [Duke Math. J. 67, No. 2, 423-459 (1992; Zbl 0777.53006)]\ proved the existence of a \(C^{1,1}\) embedding if \(K(g)\) is positive everywhere except at one point \(P\), and \(\Delta_g K\geq 0\) near \(P\). Here the authors extend this result to arbitrary \(C^4\) metrics of nonnegative curvature. The same result was also obtained recently by \textit{P. F. Guan} and \textit{Y. Y. Li} [J. Differ. Geom. 39, No. 2, 331-342 (1994; Zbl 0796.53056)]\ using a somewhat different method.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 795024
zbMATH Keywords: Weyl problem
zbMATH Keywords: nonnegative Gauss curvature
zbMATH Keywords: Monge-Ampère equation
author: Jiaxing Hong
author: Claude Zuily
reviewed by: John Urbas
MaRDI profile type: Publication
cites work: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation
cites work: On the regularity of the solution of then-dimensional Minkowski problem
cites work: Monge-Ampere Equations on Convex Regions of The Plane
cites work: \(C^\infty\) convex functions and manifolds of positive curvature
cites work: Neue a-priori-Abschätzungen für den Ortsvektor einer Fläche positiver Gaußscher Krümmung durch ihr Linienelement
cites work: Q5730622
cites work: Dirichlet problems for general Monge-Ampère equations
cites work: Isometric embeddings of surfaces with nonnegative curvature in \(\mathbb{R}^ 3\)
cites work: The Weyl and Minkowski problems in differential geometry in the large
cites work: Q4069454
cites work: Q3844833
cites work: Regularity of Solutions of Two-Dimensional Monge-Ampere Equations
OpenAlex ID: W2002815184
description: scientific article; zbMATH DE number 795024