2023/02/13 by Oh, Yong-Geun, Kim, Taesu
#53C07 (Secondary) #53D42 (Primary) 58J32 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2302.06122
In this survey article, we present the analysis of pseudoholomorphic curves u:( Σ,j) → (Q × ℝ, \widetilde J) on the symplectization of contact manifold (Q,λ) as a subcase of the analysis of contact instantons w: Σ→ Q, i.e., of the maps w satisfying the equation ∂πw = 0, d(w^*λ∘ j) = 0 on the contact manifold (Q,λ), which has been carried out by a coordinate-free covariant tensorial calculus. When the analysis is applied to that of pseudoholomorphic curves u = (w,f) with w = πQ ∘ u, f = s∘ u on symplectization, the outcome is generally stronger and more accurate than the common results on the regularity presented in the literature in that all of our a priori estimates can be written purely in terms w not involving f. The a priori elliptic estimates for w are largely consequences of various Weitzenböck-type formulae with respect to the contact triad connection introduced by Wang and the first author in [OW14], and the estimate for f is a consequence thereof by simple integration of the equation df = w^*λ∘ j. We also derive a simple precise tensorial formulae for the linearized operator and for the asymptotic operator that admit a perturbation theory of the operators with respect to (adapted) almost complex structures: The latter has been missing in the analysis of pseudoholomorphic curves on symplectization in the existing literature.