The Jacobi operator of some special minimal hypersurfaces

TL;DR

This paper investigates the stability and nondegeneracy of certain minimal hypersurfaces in Euclidean space, focusing on their Jacobi operators, with results depending on the dimension sum being above or below 8.

Contribution

It provides a classification of Jacobi fields and stability properties for special minimal hypersurfaces asymptotic to Lawson cones, extending understanding of their geometric analysis.

Findings

Hypersurfaces with m+n≥8 are strictly stable.

Full classification of bounded Jacobi fields for these hypersurfaces.

Hypersurfaces with m+n≤7 have infinite Morse index.

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 $m,n\geq 2$. 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.