[Paper Review] Stability of closed characteristics on compact convex hypersurfaces in $\R^6$

Wei Wang|ArXiv.org|Jan 24, 2007

Geometry and complex manifolds11 citations

TL;DR

This paper investigates the stability and spectral properties of closed characteristics on compact convex hypersurfaces in $\mathbb{R}^6$. Using Morse theory and mean index identities for Hamiltonian systems, it proves that if such a hypersurface $\Sigma$ has finitely many geometrically distinct closed characteristics, then at least two must have irrational mean indices, and if exactly three exist, at least two must be elliptic.

ABSTRACT

In this paper, let $Σ\subset\R^{6}$ be a compact convex hypersurface. We prove that if $Σ$ carries only finitely many geometrically distinct closed characteristics, then at least two of them must possess irrational mean indices. Moreover, if $\Sg$ carries exactly three geometrically distinct closed characteristics, then at least two of them must be elliptic.

Motivation & Objective

• To establish stability and spectral properties of closed characteristics on compact convex hypersurfaces in $\mathbb{R}^6$.
• To investigate the distribution of mean indices and ellipticity when only finitely many geometrically distinct closed characteristics exist.
• To extend prior results on non-degeneracy and stability in symplectic geometry to the six-dimensional case.
• To prove that under finitely many closed characteristics, at least two must have irrational mean indices.
• To show that if exactly three geometrically distinct closed characteristics exist, at least two must be elliptic.

Proposed method

• Utilizes the Hamiltonian system defined by $H_\alpha(x) = j(x)^\alpha$ for $\alpha \in (1,2)$, where $j$ is the gauge function of the hypersurface $\Sigma$.
• Applies Morse theory on the free loop space with $S^1$-action to compute critical modules $C_{S^1,q}(\Psi_a, S^1 \cdot u)$ associated with closed characteristics.
• Employs the mean index identity and the $\hat{\chi}$-invariant to relate the Conley-Zehnder indices of iterates to the topology of the loop space.
• Uses the $S^1$-equivariant Morse theory and the $\hat{\chi}$-invariant to derive contradictions in cases where assumptions on ellipticity or index parity fail.
• Applies the Bott-type iteration formula for Conley-Zehnder indices to analyze the parity and growth of $i(y^m)$ for iterates $y^m$.
• Relies on the symplectic path $\gamma_y(t)$ associated with each closed characteristic and its Floquet multipliers to determine ellipticity and non-degeneracy.

Experimental results

Research questions

• RQ1 What is the minimal number of closed characteristics with irrational mean indices on a compact convex hypersurface in $\mathbb{R}^6$ with finitely many geometrically distinct closed characteristics?
• RQ2 Under what conditions must at least two closed characteristics on such a hypersurface be elliptic?
• RQ3 Can the $\hat{\chi}$-invariant and mean index identity be used to rule out specific configurations of closed characteristics in $\mathbb{R}^6$?
• RQ4 How does the $S^1$-equivariant Morse theory constrain the possible Conley-Zehnder index patterns of closed characteristics?
• RQ5 What topological and spectral obstructions arise when assuming only three geometrically distinct closed characteristics exist?

Key findings

• If $\Sigma \in \mathcal{H}(6)$ has finitely many geometrically distinct closed characteristics, then at least two must possess irrational mean indices.
• If $\Sigma \in \mathcal{H}(6)$ has exactly three geometrically distinct closed characteristics, then at least two of them must be elliptic.
• The proof relies on contradiction via the $\hat{\chi}$-invariant and the mean index identity, showing that assumed index patterns lead to $\hat{\chi}$-sums less than $1/2$, violating the identity.
• The Conley-Zehnder indices of iterates satisfy specific parity conditions: $i(y_j^m) \in 2\mathbb{N}$ for odd $m$, and $i(y_j^m) \in 2\mathbb{N}_0 + 1$ for even $m$, ensuring non-degeneracy.
• The critical modules $C_{S^1,q}(\Psi_a, S^1 \cdot u_j^m)$ are computed via Morse theory, and their dimensions are used to derive the $M_q = b_q$ equality, which is essential for contradiction.
• In all cases, the assumption of only three closed characteristics leads to a $\hat{\chi}$-sum strictly less than $1/2$, contradicting the known identity $\sum \frac{\hat{\chi}(y_j)}{\hat{i}(y_j)} = \frac{1}{2}$, thus proving the result.