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

TL;DR

This paper proves the existence of at least three elliptic closed characteristics on certain non-degenerate convex hypersurfaces in six-dimensional space, advancing understanding of their stability and symplectic properties.

Contribution

It establishes the existence of at least two elliptic closed characteristics in general, and at least three in the non-degenerate case in R^6, supporting the three elliptic characteristics conjecture.

Findings

At least two elliptic closed characteristics exist on such hypersurfaces.

In non-degenerate cases, at least three elliptic characteristics are guaranteed.

Two of these elliptic characteristics are irrationally elliptic.

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.