Analysis of contact Cauchy-Riemann maps. II: Canonical neighborhoods and exponential convergence for the Morse-Bott case. (English) Zbl 1410.53080
Summary: This is a sequel to our papers [in: Real and complex submanifolds. Proceedings of the ICM 2014 satellite conference and of the 18th international workshop on differential geometry, Daejeon, Korea, August 10–12, 2014. Tokyo: Springer. 43–63 (2014; Zbl 1323.53086); J. Math. 55, No. 4, 647–679 (2018; Zbl 1454.53074)]. In [loc. cit., 2014], we introduced a canonical affine connection on \(M\) associated to the contact triad \((M,\lambda,J)\). In [loc. cit., 2018], we used the connection to establish a priori \(W^{k,p}\)-coercive estimates for maps \(w:\dot{\Sigma}\rightarrow M\) satisfying \(\overline{\partial^{\pi}} w=0\), \(d(w^\ast\lambda\circ j)=0\) 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 F. Bourgeois [A Morse-Bott approach to contact homology. Stanford: Stanford University (PhD Thesis) (2002)] (resp. by E. Bao [Pac. J. Math. 278, No. 2, 291–324 (2015; Zbl 1327.53110)]), 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).
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 F. Bourgeois [A Morse-Bott approach to contact homology. Stanford: Stanford University (PhD Thesis) (2002)] (resp. by E. Bao [Pac. J. Math. 278, No. 2, 291–324 (2015; Zbl 1327.53110)]), 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).
MSC:
| 53D10 | Contact manifolds (general theory) |
| 32V05 | CR structures, CR operators, and generalizations |
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