[Paper Review] Closed characteristics on compact convex hypersurfaces in $\R^{2n}$

Yiming Long, Chaofeng Zhu|ArXiv.org|Sep 18, 2001

Geometric Analysis and Curvature Flows21 references3 citations

TL;DR

This paper establishes the existence of at least $\varrho_n(\Sigma) \geq \lfloor n/2 \rfloor + 1$ geometrically distinct closed characteristics on any $C^2$ compact convex hypersurface $\Sigma \subset \mathbb{R}^{2n}$ with nonempty interior. It proves that if all closed characteristics are nondegenerate, then $\varrho_n(\Sigma) \geq n$, and if the total number is finite, there exists at least one elliptic closed characteristic and $\varrho_n(\Sigma) - 1$ with irrational mean indices, with at least two elliptic ones if the total is at most $2\varrho_n(\Sigma) - 2$. The results are derived via symplectic topology and index theory on the Hamiltonian flow on the energy level set $H_\alpha^{-1}(1)$. The key contribution is a new invariant $\varrho_n(\Sigma)$ that controls the minimal number of geometrically distinct closed characteristics and their dynamical properties.

ABSTRACT

For any given compact C^2 hypersurface Σin {\bf R}^{2n} bounding a strictly convex set with nonempty interior, in this paper an invariant \varrho_n(Σ) is defined and satisfies \varrho_n(Σ)\ge [n/2]+1, where [a] denotes the greatest integer which is not greater than a\in {\bf R}. The following results are proved in this paper. There always exist at least ρ_n(Σ) geometrically distinct closed characteristics on Σ. If all the geometrically distinct closed characteristics on Σare nondegenerate, then \varrho_n(Σ)\ge n. If the total number of geometrically distinct closed characteristics on Σis finite, there exists at least an elliptic one among them, and there exist at least \varrho_n(Σ)-1 of them possessing irrational mean indices. If this total number is at most 2\varrho_n(Σ) -2, there exist at least two elliptic ones among them.

Motivation & Objective

• To establish a lower bound on the number of geometrically distinct closed characteristics on compact $C^2$ convex hypersurfaces in $\mathbb{R}^{2n}$.
• To define and analyze a new symplectic invariant $\varrho_n(\Sigma)$ that controls the minimal number of such closed characteristics.
• To investigate the dynamical properties—such as nondegeneracy, ellipticity, and irrational mean indices—of closed characteristics under finiteness assumptions.
• To generalize and refine earlier results on closed characteristics, including the existence of at least two elliptic ones when the total count is small.

Proposed method

• Define the gauge function $j_C(x)$ for the convex body $C$ bounded by $\Sigma$, and construct the Hamiltonian $H_\alpha(x) = j_C(x)^\alpha$ for $\alpha \in (1,2)$, so that $\Sigma = H_\alpha^{-1}(1)$.
• Use the Hamiltonian system $\dot{x} = J H_\alpha'(x)$ with $H_\alpha(x) = 1$ to model closed characteristics as $1$-periodic solutions of the flow.
• Apply symplectic topological methods, particularly the Maslov-type index theory and the $S^1$-equivariant cohomology (Fadell-Rabinowitz index), to analyze the critical points of the action functional on the loop space.
• Define the invariant $\varrho_n(\Sigma)$ via the splitting numbers and ultimate types of Floquet multipliers of the associated symplectic paths, using the $\diamond$-product decomposition of symplectic matrices.
• Use the spectral flow and the index iteration formula to relate the Morse-type index $i(\gamma, m)$ and nullity $\nu(\gamma, m)$ of iterated closed characteristics to the topological structure of the monodromy matrix $\gamma(\tau)$.
• Leverage the Fadell-Rabinowitz $S^1$-cohomological index to control the category of critical orbits and derive lower bounds on the number of geometrically distinct closed characteristics.

Experimental results

Research questions

• RQ1 What is the minimal number of geometrically distinct closed characteristics on any compact $C^2$ convex hypersurface $\Sigma \subset \mathbb{R}^{2n}$?
• RQ2 How does the new invariant $\varrho_n(\Sigma)$ relate to the number and dynamical types (elliptic, nondegenerate, hyperbolic) of closed characteristics on $\Sigma$?
• RQ3 Under what conditions does the finiteness of the number of geometrically distinct closed characteristics imply the existence of at least one elliptic one?
• RQ4 Can the number of closed characteristics with irrational mean indices be bounded below in terms of $\varrho_n(\Sigma)$?
• RQ5 What is the minimal number of elliptic closed characteristics when the total number is bounded above by $2\varrho_n(\Sigma) - 2$?

Key findings

• For any $\Sigma \in \mathcal{H}(2n)$, there exist at least $\varrho_n(\Sigma) \geq \lfloor n/2 \rfloor + 1$ geometrically distinct closed characteristics.
• If all geometrically distinct closed characteristics on $\Sigma$ are nondegenerate, then $\varrho_n(\Sigma) \geq n$, which improves the lower bound.
• If the total number of geometrically distinct closed characteristics is finite, then at least one is elliptic, and at least $\varrho_n(\Sigma) - 1$ have irrational mean indices.
• If the total number of geometrically distinct closed characteristics is at most $2\varrho_n(\Sigma) - 2$, then there are at least two elliptic ones.
• The invariant $\varrho_n(\Sigma)$ is defined via the splitting numbers and ultimate types of Floquet multipliers of the symplectic path associated to each closed characteristic, and it is invariant under $C^1$-perturbations of the hypersurface.
• The results are derived using the $S^1$-equivariant Fadell-Rabinowitz cohomological index and the index iteration formula, which relate the topological complexity of the loop space to the spectral data of the monodromy matrix.