Pseudoholomorphic curves on the LCS-fication of contact manifolds

Abstract

Abstract For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S1 = Mid (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form ∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ $$\bar{\partial}^{\pi} w=0, \quad w^{*} \lambda \circ j=f^{*} d \theta$$ for the map u = (w, f) : Σ˙→Q×S1$\dot{\Sigma} \rightarrow Q \times S^{1}$for a λ-compatible almost complex structure J and a punctured Riemann surface (Σ˙,j).$(\dot{\Sigma}, j).$In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H1(Σ˙,Z)$H^{1}(\dot{\Sigma}, \mathbb{Z})$and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of (Q, λ) (more generally on arbitrary locally conformal symplectic manifolds).