The existence of two closed characteristics on every compact star-shaped hypersurface in ${\bf R}^4$

Abstract

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in ${\bf R}^4$. 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.

To appear in Acta Mathematica Sinica, English Series (Special issue for the 29th anniversary of the establishment of the Shiing-Shen Chern Prize of the Chinese Mathematical Society). This is the final version. arXiv admin note: substantial text overlap with arXiv:1308.3543; and text overlap with arXiv:math/0701673 by other authors