[Paper Review] Equivariant symplectic homology and multiple closed Reeb orbits

Jungsoo Kang|arXiv (Cornell University)|May 6, 2013

Geometric and Algebraic Topology44 references3 citations

TL;DR

This paper establishes the existence of multiple closed Reeb orbits on certain contact manifolds using $S^1$-equivariant symplectic homology and the Conley-Zehnder index iteration formula. It proves that nondegenerate contact manifolds admitting displaceable exact contact embeddings into convex-at-infinity symplectic manifolds with trivial first Chern class possess at least two contractible closed Reeb orbits under homological conditions, and extends this to 3- and 5-dimensional cases with quantitative lower bounds based on Betti numbers.

ABSTRACT

We study the existence of multiple closed Reeb orbits on some contact manifolds by means of $S^1$-equivariant symplectic homology and the index iteration formula. It is proved that a certain class of contact manifolds which admit displaceable exact contact embeddings, a certain class of prequantization bundles, and Brieskorn spheres have multiple closed Reeb orbits.

Motivation & Objective

• To establish the existence of multiple closed Reeb orbits on contact manifolds that admit displaceable exact contact embeddings.
• To extend multiplicity results beyond 3-dimensional contact manifolds to higher dimensions using equivariant symplectic homology.
• To provide quantitative lower bounds on the number of closed Reeb orbits based on Betti numbers of filling domains.
• To investigate whether subcritical Weinstein fillable contact manifolds always have at least two closed Reeb orbits.
• To explore the applicability of the method to prequantization bundles and Brieskorn spheres.

Proposed method

• Utilizes $S^1$-equivariant symplectic homology to analyze the topology of the contact manifold and its filling.
• Applies the Conley-Zehnder index iteration formula to study the behavior of multiple covers of Reeb orbits.
• Employs homological conditions on the filling domain $W_0$—specifically $H_*(W_0, \Sigma; \mathbb{Q})$ in certain degrees—to derive non-vanishing homology classes.
• Imposes the condition $c_1(W)|_{\pi_2(W)} = 0$ to ensure compatibility with the equivariant homology computation.
• Uses the fact that $\alpha - \lambda|_\Sigma$ is exact to relate the contact form to the symplectic structure on $W$, enabling the use of Hamiltonian dynamics.
• Analyzes the Conley-Zehnder index growth for elliptic, hyperbolic, and non-elliptic non-hyperbolic orbits to derive contradictions under the assumption of a single simple orbit.

Experimental results

Research questions

• RQ1 Does every nondegenerate subcritical Weinstein fillable closed contact manifold have at least two closed Reeb orbits?
• RQ2 Can the method of $S^1$-equivariant symplectic homology and index iteration detect multiple closed Reeb orbits in higher-dimensional contact manifolds beyond 3-manifolds?
• RQ3 What is the minimal homological condition on the filling domain $W_0$ that guarantees multiple closed Reeb orbits?
• RQ4 Are there general conditions under which a contact manifold with a displaceable exact contact embedding must have more than two closed Reeb orbits?
• RQ5 How do Betti numbers of the filling domain relate to the number of closed Reeb orbits in the contact boundary?

Key findings

• For a 3-dimensional closed contact manifold with a displaceable exact contact embedding into a convex-at-infinity symplectic manifold with $c_1(W)|_{\pi_2(W)} = 0$, the number of contractible closed Reeb orbits is at least $b_3(W_0, \Sigma; \mathbb{Q}) + 2$.
• In the case of a 3-manifold, there are at least $b_2(\Sigma; \mathbb{Q}) + 2$ closed Reeb orbits if $W$ is subcritical Weinstein.
• For a 5-dimensional contact manifold with $b_2(W_0, \Sigma; \mathbb{Q}) = 1$ and $b_4(W_0, \Sigma; \mathbb{Q}) \geq 4$, there are at least two closed Reeb orbits contractible in $W$.
• The Conley-Zehnder index of $b_3(W_0, \Sigma; \mathbb{Q})$-many simple closed Reeb orbits is exactly 2 in the 3-dimensional case.
• The assumption of a single simple closed Reeb orbit leads to a contradiction in both 3- and 5-dimensional cases due to irrationality constraints in the index iteration formula.
• The method applies to rational homology spheres, $\pi_1$-injective fillable 5-manifolds, and subcritical Weinstein fillable 7-manifolds, all of which satisfy condition (ii) in Theorem A.