Lawson cones and the Allen-Cahn equation

TL;DR

This paper investigates the stability of Lawson cones and constructs complex solutions to the Allen-Cahn equation in high dimensions, whose zero level sets have multiple connected components and infinite Morse index.

Contribution

It establishes nondegeneracy and stability properties of Lawson cones and uses them to build multi-component solutions to the Allen-Cahn equation with infinite Morse index.

Findings

Stability properties of Lawson cones are characterized.

Construction of multi-component solutions to the Allen-Cahn equation.

Solutions have zero level sets with multiple connected components and infinite Morse index.

Abstract

In this paper we discuss nondegeneracy and stability properties of some special minimal hypersurfaces which are asymptotic to a given Lawson cone $C_{m,n}$, for $m,\,n\ge 2$. Then we use such hypersurfaces to construct solutions to the Allen-Cahn equation $-\Delta u=u-u^3$ in $\R^{N+1}$, $N+1\ge 8$, whose zero level set has exactly $k\ge 2$ connected components and with infinite Morse index.