2013/11/25 by Oh, Yong-Geun, Wang, Rui
#53D42 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1311.6196
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,λ,J). In [OW2], they used the connection to establish a priori Wk,p-coercive estimates for maps w: Σ→ M satisfying ∂πw= 0, d(w^*λ∘ 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,λ,J), with λ 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).