[Paper Review] The asymptotics of ECH capacities

Daniel Cristofaro‐Gardiner, Michael Hutchings+1 more|arXiv (Cornell University)|Oct 8, 2012

Geometric and Algebraic Topology12 references4 citations

TL;DR

This paper establishes that the asymptotic growth rate of Embedded Contact Homology (ECH) capacities for a four-dimensional Liouville domain with finite capacities recovers the symplectic volume. Using a deep connection between ECH and Seiberg-Witten Floer cohomology, the authors prove that the limit of $ c_k^2 / k $ as $ k \to \infty $ equals $ 4 \cdot \text{vol}(X,\omega) $, confirming a conjecture and providing a dynamical refinement of the Weinstein conjecture.

ABSTRACT

In a previous paper, the second author used embedded contact homology (ECH) of contact three-manifolds to define "ECH capacities" of four-dimensional symplectic manifolds. In the present paper we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This follows from a more general theorem relating the volume of a contact three-manifold to the asymptotics of the amount of symplectic action needed to represent certain classes in ECH. The latter theorem was used by the first and second authors to show that every contact form on a closed three-manifold has at least two embedded Reeb orbits.

Motivation & Objective

• To prove that the asymptotic growth of ECH capacities recovers the symplectic volume of a four-dimensional Liouville domain.
• To establish a general theorem linking the volume of a contact three-manifold to the asymptotics of symplectic action in ECH.
• To provide a dynamical refinement of the Weinstein conjecture by showing every nondegenerate contact form on a closed three-manifold has at least two embedded Reeb orbits.
• To extend prior results on star-shaped domains in $\mathbb{R}^4$ to general Liouville domains with finite ECH capacities.

Proposed method

• Utilizes embedded contact homology (ECH) to define a sequence of symplectic invariants called ECH capacities.
• Applies Taubes' isomorphism between ECH and Seiberg-Witten Floer cohomology to translate ECH computations into topological invariants.
• Constructs a cobordism from a contact manifold to a disjoint union of contact balls and uses the cobordism map on ECH to relate action functionals.
• Employs the $U$-map in ECH to analyze the structure of chain complex generators and their action under iteration.
• Uses the known asymptotic behavior of ECH capacities on standard balls $B(r)$, where $ c_{\zeta_k}(\partial B(r)) = dr $ with $ d \approx \sqrt{2k} $, to derive volume recovery.
• Combines these tools via a limiting argument over increasing action levels, leveraging the composition property of cobordism maps and the structure of the $U$-action on homology.

Experimental results

Research questions

• RQ1 Does the asymptotic growth of ECH capacities recover the symplectic volume of a four-dimensional Liouville domain?
• RQ2 Can the volume of a contact three-manifold be recovered from the asymptotics of symplectic action in ECH?
• RQ3 What is the precise asymptotic behavior of $ c_k(X,\omega)^2 / k $ as $ k \to \infty $ for domains with finite ECH capacities?
• RQ4 How does the $U$-map structure in ECH constrain the action growth of homology classes?
• RQ5 Can the asymptotic ECH capacity behavior be used to strengthen the Weinstein conjecture?

Key findings

• The asymptotic growth of ECH capacities satisfies $ \lim_{k\to\infty} \frac{c_k(X,\omega)^2}{k} = 4 \cdot \text{vol}(X,\omega) $ for any four-dimensional Liouville domain with all $ c_k < \infty $.
• The result confirms a conjecture from [4, Conj. 1.12] and extends prior results on star-shaped domains in $\mathbb{R}^4$.
• For the standard ball $ B(r) $, the ECH capacity satisfies $ c_{\zeta_k}(\partial B(r)) = dr $ where $ d $ satisfies $ \frac{d^2 + d}{2} \leq k \leq \frac{d^2 + 3d}{2} $, leading to $ \lim_{k\to\infty} \frac{c_{\zeta_k}(\partial B(r))^2}{k} = 4 \cdot \text{vol}(B(r)) $.
• The proof relies on a cobordism construction from a contact manifold to a union of contact balls, using the cobordism map on ECH to relate action functionals.
• The $U$-map structure ensures that the action of iterated $U$-images controls the growth, enabling the asymptotic volume recovery.
• The result implies that the embedding obstruction from ECH capacities becomes sharp in the large $k$ limit, recovering the volume constraint $ \text{vol}(X,\omega) \leq \text{vol}(X',\omega') $.