An elementary alternative to ECH capacities

Abstract

The ECH capacities are a sequence of numerical invariants of symplectic four-manifolds which give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg-Witten theory. In this note we define a new sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The new capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The new capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff-Siegel giving a similar elementary alternative to symplectic capacities from rational SFT.

An elementary alternative to PFH spectral invariants

“…The Weyl law requires one additional ingredient, namely the asymptotics of Hutchings' elementary capacities c H k for the ball. The proof of the asymptotic formula for c H k given in [16] is elementary in the sense that it does not require ingredients beyond Gromov's paper [12], Taubes local version of Gromov compactness [29], the relative relative adjunction formula and the writhe bound [15]. We emphasize that as soon as finiteness of the spectral invariants c d is established, the Weyl law for c d is elementary in the same sense.…”

“…Motivation and background. This paper is motivated by recent work of McDuff-Siegel [21] and Hutchings [16]. Both papers provide elementary ersatz definitions of certain sequences of symplectic capacities and show that important applications of the original capacities are recovered by the alternative ones, thus leading to much more elementary proofs.…”

“…Instead of symplectic capacities, we focus on spectral invariants of area-preserving surface diffeomorphisms. Following the philosophy of [21] and [16], we define an elementary replacement for spectral invariants coming from periodic Floer homology (PFH), whose construction is highly non-trivial and in particular relies on gauge theory. PFH spectral invariants have recently found remarkable applications to C 0 symplectic geometry, Hofer geometry and symplectic dynamics in dimension two (see [5], [6], [8], [9]).…”

“…Let us begin with a brief review of Hutchings' elementary capacities defined in [16]. Recall that a symplectic capacity c is a function that assigns numbers c(X, ω) ∈ [0, ∞] to symplectic manifolds (X, ω).…”

“…Continuity is an easy consequence of the shift property, non-negativity and monotonicity. The proofs of monotonicity, the packing property and the closed curve upper bound are based on a neck stretching argument similar to the one explained in [16]. The main ingredients are a local version of Gromov compactness due to Taubes (see [29,Proposition 3.3]) and a relative adjunction formula and asymptotic writhe bound for pseudo-holomorphic curves in symplectic 4-manifolds with cylindrical ends due to Hutchings (see [15,Proposition 4.9 and Lemma 4.20]).…”