Three closed characteristics on non-degenerate star-shaped hypersurfaces in ${\mathbf{R}}^6$

In this paper, we prove that for every non-degenerate $C^3$ compact star-shaped hypersurface $\mathrm{\Sigma }$ in ${\mathbf{R}}^6$ which carries no prime closed characteristic of Maslov-type index $0$ or no prime closed characteristic of Maslov-type index $-1$, there exist at least three prime closed characteristics on $\mathrm{\Sigma }$.