Four closed characteristics on compact star-shaped hypersurfaces in $\mathbb{R}^{8}$

In this paper, we proved that for every non-degenerate $C^3$ compact star-shaped hypersurface $\Sigma$ in $\mathbb{R}^{8}$ which carries no prime closed characteristic of Maslov-type index $-1$, there exist at least four prime closed characteristics on $\Sigma$.