Pseudoholomorphic curves on the LCS-fication of contact manifolds

Abstract

For each contact diffeomorphism phi : (Q, xi). (Q, xi) of ( Q, xi), we equip its mapping torus M-phi with a locally conformal symplectic form of Banyaga's type, which we call the lcs mapping torus of the contact diffeomorphism phi. In the present paper, we consider the product Q x S-1 = M-id (corresponding to phi = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form partial derivative(pi)w = 0, w*lambda degrees j = f * d theta for the map u = (w, f) : (Sigma) over dot -> Q x S-1 for a lambda-compatible almost complex structure J and a punctured Riemann surface ((Sigma) over dot, 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 H-1((Sigma) over dot, 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, lambda) (more generally on arbitrary locally conformal symplectic manifolds).