Submanifolds with constant Jordan angles and rigidity of the Lawson-Osserman cone
type: Item
instance of: scholarly article
title: Submanifolds with constant Jordan angles and rigidity of the Lawson-Osserman cone
zbMATH Open document ID: 1393.53010
author: Ling Yang
author: Jürgen Jost
author: Yuanlong Xin
published in: The Asian Journal of Mathematics
publication date: 2018-05-16T00:00:00Z
review text: The Lawson-Osserman cone is a four-dimensional cone in the imaginary of octonions \(\text{Im }{\mathbb Q}\), which is the graph of a cone-like function. This gives a basic counterexample for Bernstein-type results for minimal graphs of higher codimension in Euclidean space. While this cone-like function giving the Lawson-Osserman cone is an entire solution of the minimal surface system \({\mathbb R}^4 \to {\mathbb R}^3\) that is not smooth at the origin, \textit{W. Ding} and \textit{Y. Yuan} [J. Partial Differ. Equations 19, No. 3, 218--231 (2006; Zbl 1101.49030)] constructed a family of \(4\)-dimensional entire minimal graphs in \(\text{Im }{\mathbb Q}\) that the tangent cone at infinity of each one is just the Lawson-Osserman cone. Thus, in contrast to the case of codimension \(1\), where by a theorem of \textit{J. Moser} [Commun. Pure Appl. Math. 14, 577--591 (1961; Zbl 0111.09302)] every entire minimal graph with bounded gradient is planar, these constructions show that in higher codimension, nontrivial such entire minimal graphs exist. So understanding the Lawson-Osserman cone is a key for the analysis of the higher-codimension Bernstein problem. In this paper, the authors study the geometry of the Lawson-Osserman cone in terms of its basic constant Jordan angle and show a rigidity result on the higher-codimension Bernstein problem. Let \(M^n\) be an \(n\)-dimensional submanifold in \({\mathbb R}^{n+m}\) and \(Q_0\) be a fixed \(m\)-plane. For each tangent space \(T_pM\), the tangent Jordan angle at \(p\) between \(T_pM\) and \(Q_0\) are critical values of the angle between a nonzero vector in \(T_pM\) and its orthogonal projection in \(Q_0\) as tangent vectors run through \(T_pM\), and the normal Jordan angle at \(p\) is defined simularly. If the normal Jordan angle is independent of the point \(p\) and the multiplicity of each normal Jordan angle is constant, \(M\) is said to have constant Jordan angle (CJA) relative to \(Q_0\). The authors prove that if \(f\…
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6868969
zbMATH Keywords: Lawson-Osserman cone
zbMATH Keywords: constant Jordan angle
zbMATH Keywords: coassociative submanifold
zbMATH Keywords: minimal graph
zbMATH Keywords: Bernstein problem
MaRDI profile type: Publication
full work available at URL: ajm.2018.v22.n1.a3
OpenAlex ID: W2802883163
Wikidata QID: Q129850898
DOI: 10.4310/AJM.2018.V22.N1.A3
Recommended article: RIGIDITY AND SPHERE THEOREMS FOR SUBMANIFOLDS
Recommended article: Q4946601
Recommended article: Q3834654
Recommended article: Rigidity theorems for submanifolds
Recommended article: Rigidity of submanifolds in the Euclidean and spherical spaces
Recommended article: Q3373506
Recommended article: An optimal rigidity theorem for complete submanifolds in a sphere
Recommended article: Rigidity and sphere theorems for submanifolds. II
Recommended article: Q3802199
Recommended article: Geometric and differentiable rigidity of submanifolds in spheres
description: scientific article; zbMATH DE number 6868969