Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lu Liu, Yuwei Ou

Abstract

Let $Σ\subset \mathbb{R}^{2n}$ 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 $Σ$ 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 $Σ\subset \mathbb{R}^{2n}$ 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 $Σ$ is non-degenerate. Moreover, for any non-degenerate $Σ\subset\mathbb{R}^{6}$ 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.

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

Abstract

Let with be any 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 possesses only finitely geometrically distinct closed characteristics. This conjecture has been fully resolved only in , while it remains completely open in higher dimensions. Even in , it is unknown whether there exist three elliptic closed characteristics. In this paper, we first prove that for any 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 is non-degenerate. Moreover, for any non-degenerate with finitely many closed characteristics, we obtain at least three elliptic characteristics, of which at least two are irrationally elliptic. Based on the -or- conjecture, three elliptic closed characteristics are optimal. This result provide theoretical support for further research on this conjecture.
Paper Structure (6 sections, 14 theorems, 72 equations)

This paper contains 6 sections, 14 theorems, 72 equations.

Table of Contents

  1. Introduction and main results
  2. Review of the Maslov-type index theory
  3. Definition of the Maslov-type index
  4. Iterative formula of the Maslov-type index
  5. Common index jump theorem
  6. The elliptic closed characteristics

Key Result

Theorem 1.2

For any surfaces $\Sigma \in \mathcal{H}(2n)$ with $^{\#}\widetilde{\mathcal{J}}(\Sigma) < +\infty$, then at least two closed characteristics are elliptic whose momodormy matrix possess a nice symplectic normal form with all $N_{2}(\lambda_{1},\nu_{1}),\ldots,N_{2}(\lambda_{r_{0}},\nu_{r_{0}})$ are rational normal form. Moreover, when $\Sigma$ is non-degenerate, then they are irrationally ellipti

Theorems & Definitions (28)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • ...and 18 more