Stability of the saddle solutions for the Allen-Cahn equation

We are concerned with the saddle solutions of the Allen-Cahn equation constructed by Cabr\'{e} and Terra \cite{C,C2} in $\mathbb{R}^{2m}$. These solutions vanish precisely on the Simons cone. The existence and uniqueness of saddle solution are shown in \cite{C,C2,C1}. Regarding the stability, Schatzman \cite{Sch} proved that the saddle solution is unstable for $m=1,$ Cabr\'{e} \cite{C1} showed the instability for $m=2,3$ and stability for $m\geq7$. This has left open the case of $m=4,5,6$. In this paper we show that the saddle solutions are stable when $m=4,5,6$, thereby confirming Cabr\'{e}'s conjecture in \cite{C1}. The conjecture that saddle solutions in dimensions $2m\geq8$ should be global minimizers of the energy functional remains open.