Instability of type (II) Lawson-Osserman cones (Q6619977)

The celebrated paper by \textit{H. B. Lawson jun.} and \textit{R. Osserman} [Acta Math. 139, 1--17 (1977; Zbl 0376.49016)] has been object of numerous investigation. In the paper by \textit{X. Xu} et al. [J. Math. Pures Appl. (9) 129, 266--300 (2019; Zbl 1425.53011)], Lawson-Osserman cones in Euclidean spaces were classified into two categories: Type (I) and Type (II), see the above paper for details, as well as the recent preprint by \textit{Y. Zhang} [``On the non-existence of solutions of the Dirichlet problem for the minimal surface system'', Preprint, \url{arXiv:1812.11553}]. In the paper under review the authors prove that all those of Type (II) are unstable (Theorem 1.1) and certain Lawson-Osserman rays of those of Type (I) are stable (Theorem 1.2). These theorems are obtained through a classic Jacobi field analysis, while relating them to a dynamic system stability behaviour, hence the authors assure the existence of an infinite family of non-smooth unstable minimal graphs. These results provide a remarkable contribution to the literature, originated with Lawson and Osserman approach, see e.g., other papers by Y. S. Zhang and collaborators on the subject such as the ones quoted in the paper references.