The Jacobi operator of some special minimal hypersurfaces

In this work we discuss stability and nondegeneracy properties of some special families of minimal hypersurfaces embedded in \(\mathbb {R}^m\times \mathbb {R}^n\) ${\mathbb{R}}^m\times {\mathbb{R}}^n$ with \(m,n\ge 2\) $m,n\ge 2$ . These hypersurfaces are asymptotic at infinity to a fixed Lawson cone \(C_{m,n}\) $C_{m,n}$ . In the case \(m+n\ge 8\) $m+n\ge 8$ , we show that such hypersurfaces are strictly stable and we provide a full classification of their bounded Jacobi fields, which in turn allows us to prove the non-degeneracy of such surfaces. In the case \(m+n\le 7\) $m+n\le 7$ , we prove that such hypersurfaces have infinite Morse index.