The Jacobi operator of some special minimal hypersurfaces

Abstract

In this work we discuss stability and nondegeneracy properties of some special families of minimal hypersurfaces embedded in \(\mathbb {R}^m\times \mathbb {R}^n\) with \(m,n\ge 2\). These hypersurfaces are asymptotic at infinity to a fixed Lawson cone \(C_{m,n}\). In the case \(m+n\ge 8\), 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 \(m+n\le 7\), we prove that such hypersurfaces have infinite Morse index.