Integrability of Lawson-Osserman Cone and its Applications

In this paper, we characterize all eigenfunctions corresponding to nonpositive eigenvalues of the Jacobi operator of the link $M$ of the Lawson-Osserman cone $\mathbf{C}$ in ${\mathbb{R}}^7$. In particular, we prove that $\mathbf{C}$ is integrable, i.e., all Jacobi fields on $\mathbf{C}$ of homogeneous degree 1 and 0, are generated by rotations and translations in ${\mathbb{R}}^7$. As applications, we prove that $M$ is rigid as minimal submanifolds in ${\mathbb{S}}^6$, and derive the optimal decay order for minimal submanifolds in ${\mathbb{R}}^7$ asymptotic to $\mathbf{C}$ at infinity.