Lawson cones and the Allen-Cahn equation

TL;DR

The paper investigates stability and nondegeneracy of minimal hypersurfaces asymptotic to Lawson cones $C_{m,n}$ and leverages these geometries to construct entire solutions to the Allen-Cahn equation $\Delta u - F'(u)=0$ in high dimensions ($N+1\ge 8$) whose zero level sets have multiple components and infinite Morse index. It proves the existence of two symmetric, strictly stable minimal hypersurfaces $\Sigma^{\pm}_{m,n}$ for $m+n\ge 8$, and a unique, nondegenerate, infinite-Morse-index surface $\Sigma_{m,n}$ for $m+n\le 7$, with detailed analysis of the Jacobi operator $J_{\Sigma}=\Delta_{\Sigma}+|A_{\Sigma}|^2$ and associated Jacobi fields. The Allen-Cahn constructions reduce to a Jacobi-Toda system on $\Sigma$, solved via Lyapunov-Schmidt reduction, yielding invariant solutions with two (and more) nodal components over $\Sigma_{\varepsilon}$ and establishing energy bounds and infinite Morse index. These results deepen the link between minimal-surface geometry and phase-field models in high dimensions, providing explicit asymptotics and nontrivial Morse-index properties.

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 $-Δu=u-u^3$ in $\R^{N+1}$, $N+1\ge 8$, whose zero level set has exactly $k\ge 2$ connected components and with infinite Morse index.

Theorems & Definitions (11)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Theorem 2.2: AR3
• Theorem 2.3: ABPRSM
• Theorem 2.4: AR3
• Definition 2.3
• Theorem 2.5
• Theorem 3.1: AKR
• Theorem 3.2: AKR