Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization

Abstract

Listen

Translate article

Abstract In this paper, we first derive the covariant tensorial formula for the asymptotic operators of contact instantons w : Σ̇ → Q and of pseudoholomorphic curves u : (Σ̇, j) → (Q × ℝ, J͠) on the symplectization of a contact manifold (Q, λ). The formula is independent of the choice of connection on the nose and exhibits explicit dependence on the compatible pair (λ, J) of a given contact triad (Q, λ, J). Then based on this, we prove that all eigenvalues of the asymptotic operator are simple for a generic choice of compatible CR almost complex structure J for a given contact form λ. This perturbation theory has been missing in the study of pseudoholomorphic curves on symplectization.