Isometric embeddings of surfaces with nonnegative curvature in \(\mathbb{R}^ 3\)
type: Item
instance of: scholarly article
title: Isometric embeddings of surfaces with nonnegative curvature in \(\mathbb{R}^ 3\)
zbMATH Open document ID: 0777.53006
DOI: 10.1215/S0012-7094-92-06717-2
author: Joseph A. Iaia
published in: Duke Mathematical Journal
publication date: 1992-12-14T00:00:00Z
review text: The main result: Let \(g\in C^ 4\) be a Riemannian metric on \(S^ 2\) with Gauss curvature \(K\) satisfying (i) \(K(P)=0\), \(K(Q)>0\) for \(Q\neq P\); (ii) \(\Delta_ gK(Q)\geq 0\) for all \(Q\in B_ r(P)\) for some \(r>0\). Then there exists a \(C^{1,1}\) isometric embedding \(X:(S^ 2,g)\to (\mathbb{R}^ 3,h)\) where \(h\) is the standard flat metric on \(\mathbb{R}^ 3\). In addition, if \(g\in C^ 5\), \(\limsup_{Q\to P}|\nabla K|^ 2/K=C<\infty\) and \(\liminf_{Q\to P}H(Q)=c>0\), then \(k_ 1\in C^{0,1}\) in a small ball about \(P\) where \(k_ 1\) is the smaller of the two principal curvatures and \(H\) is the mean curvature of the embedding \(X\).
Mathematics Subject Classification ID: 53A05
zbMATH DE Number: 78566
zbMATH Keywords: Gauss curvature
zbMATH Keywords: isometric embedding
reviewed by: Thomas Hasanis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 78566