Kenji Fukaya, State University of New York, Stony Brook, New York, Yong-Geun Oh, Institute for Basic Sciences, Pohang, Korea, and POSTECH, Pohang, Korea, Hiroshi Ohta, Nagoya University, Japan, and Kaoru Ono, Kyoto University, Japan.

• 1. Introduction2. Preliminaries3. Statement of the gluing theorem4. Proof of the gluing theorem I: Cut-off functions and weighted Sobolev norm5. Proof of the gluing theorem II: Gluing by alternating method6. Exponential decay of $T$ derivatives7. Surjectivity and injectivity of the gluing map8. Exponential decay estimate implies smoothness of coordinate changeA. Error term estimate of non-linear Cauchy-Riemann equation IB. Estimate of Parallel transport 1C. Error term estimate of non-linear Cauchy-Riemann equation IID. Estimate of Parallel transport 2E. Estimate of the non-linearity of Exponential mapF. Estimate of Parallel transport 3G. Estimate of $T$ derivative of the error term of non-linear Cauchy-Riemann equationH. Proof of Lemma