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 Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43-63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy-Riemann maps I: a priori C-k estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43-63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on M associated to the contact triad (M, lambda, J). In Oh and Wang (Analysis of contact Cauchy-Riemann maps I: a priori C-k estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori W-k&apos;P-coercive estimates for maps w : Sigma -> M satisfying<(partial derivative)overbar>(pi) w = 0, d(w*lambda o j) = 0 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 (A Morse-Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291-324)), 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 setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT) (c) 2017 Foundation Nagoya Mathematical Journal