Three elliptic closed characteristics on the non-degenerate compact convex hypersurfaces in R^6

Abstract

Let $\Sigma\subset \mathbb{R}^{2n}$ with $n\geq2$ be any $C^2$ compact convex hypersurface. The stability of closed characteristics has attracted considerable attention in related research fields. A long-standing conjecture states that all closed characteristics are irrationally elliptic, provided $\Sigma$ possesses only finitely geometrically distinct closed characteristics. This conjecture has been fully resolved only in $\mathbb{R}^4$, while it remains completely open in higher dimensions. Even in $\mathbb{R}^6$, it is unknown whether there exist three elliptic closed characteristics. In this paper, we first prove that for any $\Sigma\subset \mathbb{R}^{2n}$ with finitely many closed characteristics, there exist at least two elliptic closed characteristics, which possess a nice symplectic normal form. In particular, as a simple corollary, they are irrational elliptic when $\Sigma$ is non-degenerate. Moreover, for any non-degenerate $\Sigma\subset\mathbb{R}^{6}$ with finitely many closed characteristics, we obtain at least three elliptic characteristics, of which at least two are irrationally elliptic. Based on the $n$-or-$\infty$ conjecture, three elliptic closed characteristics are optimal. This result provide theoretical support for further research on this conjecture.