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.