The Existence of Two Closed Characteristics on Every Compact Star-shaped Hypersurface in R~4

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in R~4. Then Ginzburg, Hein, Hryniewicz,and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al.