Pseudoholomorphic curves on the LCS-fication of contact manifolds

Abstract

For each contact diffeomorphism φ: (Q, ζ) → (Q, ζ) of (Q, ζ), 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 {equation presented} for the map u = (w, f): ς˙→Q×S1 for a λ-compatible almost complex structure J and a punctured Riemann surface (ς˙,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) 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).