Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case

Abstract

This is a sequel to the papers [OW1], [OW2]. In [OW1], the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\lambda ,J)$. In [OW2], they used the connection to establish a priori $W^{k,p}$-coercive estimates for maps $w:\dot{\Sigma }\to M$ satisfying ${\overline{\partial }}^{\pi }w=0,d(w^{\ast }\lambda \circ j)=0$ \emph{without involving symplectization}. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse-Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\lambda ,J)$, with $\lambda$ a Morse-Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes. We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse-Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois [Bou] (resp. by Bao [Ba]), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse-Bott case is an essential ingredient in the set-up of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

69 pages, final version to appear in Nagoya Math J, improvement of overall presentation

References in corpus (1)

• Analysis of Contact Cauchy-Riemann maps I: a priori $C^k$ estimates and asymptotic convergence

Cited by in corpus (5)

• Analysis of Contact Cauchy-Riemann maps I: a priori $C^k$ estimates and asymptotic convergence
• Relative Invariants, Contact Geometry and Open String Invariants
• Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition
• Exponential decay of $J$-holomorphic maps with finite energy
• Pseudoholomoprhic curves on the $\mathfrak{LCS}$-fication of contact manifolds