2021/03/29 by Oh, Yong-Geun
#53D42 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2103.15376
In [OW2,OW3], the authors studied the nonlinear elliptic system ∂πw = 0, d(w^*λ∘ j) = 0 without involving symplectization for each given contact triad (Q,λ, J), and established the a priori Wk,2 elliptic estimates and proved the asymptotic (subsequence) convergence of the map w: Σ→ Q for any solution, called a contact instanton, on Σ under the hypothesis ‖w^*λ‖C0 < ∞ and dπw ∈ L2 ∩ L4. The asymptotic limit of a contact instanton is a `spiraling' instanton along a `rotating' Reeb orbit near each puncture on a punctured Riemann surface Σ. Each limiting Reeb orbit carries a `charge' arising from the integral of w^*λ∘ j. In this article, we further develop analysis of contact instantons, especially the W1,p estimate for p > 2 (or the C1-estimate), which is essential for the study of compactfication of the moduli space and the relevant Fredholm theory for contact instantons. In particular, we define a Hofer-type off-shell energy Eλ(j,w) for any pair (j,w) with a smooth map w satisfying d(w^*λ∘ j) = 0, and develop the bubbling-off analysis and prove an ε-regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).