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 $\mathrm{\Sigma }$ in ${\mathbb{R}}^8$ which carries no prime closed characteristic of Maslov-type index $-1$, there exist at least four prime closed characteristics on $\mathrm{\Sigma }$.