Pseudoholomoprhic curves on the $\mathfrak{LCS}$-fication of contact manifolds

Abstract

For each contact diffeomorphism $\varphi :(Q,\xi )\to (Q,\xi )$ of $(Q,\xi )$, we equip its mapping torus $M_{\varphi }$ with a \emph{locally conformal symplectic} form of Banyaga's type, which we call the \emph{$\text{lcs}$ mapping torus} of contact diffeomorphism $\varphi$. In the present paper, we consider the product $Q\times S^1=M_{id}$ (corresponding to $\varphi =id$) and develop basic analysis of the associated $J$-holomorphic curve equation, which has the form $${\overline{\partial }}^{\pi }w=0,w^{\ast }\lambda \circ j=f^{\ast }d\theta$$ for the map $u=(w,f):\dot{\Sigma }\to Q\times S^1$ for the $\lambda$-compatible almost complex structure $J$ and a punctured Riemann surface $(\dot{\Sigma },j)$. In particular, $w$ is a \emph{contact instanton} in the sense of [OW2, OW3]. We develop a scheme of treating the non-vanishing charge by introducing the notion of \emph{charge class} in $H^1(\dot{\Sigma },\mathbb{Z})$ 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 compactification of the moduli space on the $\mathfrak{lcs}$-fication of $(Q,\lambda )$ (more generally on arbitrary locally conformal symplectic manifolds).

55 pages. arXiv admin note: text overlap with arXiv:2103.15376;v2) 57 pages, introduction amplified, typos corrected, accepted in Adv. Geom

References in corpus (1)

• Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition