Submanifolds with constant Jordan angles and rigidity of the Lawson-Osserman cone (Q1749216)

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 \(f\) is an \({\mathbb R}^m\)-valued function on an open domain \(D \subset {\mathbb R}^n\), and \(M = \mathrm{Graph}(f)\) is a minimal submanifold with CAJ relative to \({\mathbb R}^m\) such that \(g^N , g^T \leq 2\), then \(M\) has to be affine \(n\)-plane. Here \(g^N\) and \(g^T \) denote the numbers of distinct normal Jordan angles and tangent Jordan angles, respectively. The second main result in this paper is the following. Let \(f\) be a smooth function from an open domain \(D \subset {\mathbb H}^n\) into \(\text{Im } {\mathbb H}\). If \(M = \mathrm{Graph}(f)\) is a coassociative submanifold with CAJ relative to \(\text{Im }{\mathbb H}\) with \(g^N \leq 2\) and \(g^T \leq 3\), then either \(M\) is an affine \(4\)-plane or a translate of an open subset of the Lawson-Osserman cone. Here \({\mathbb H}\) denotes the quaternions.