Geometric analysis of perturbed contact instantons with Legendrian boundary conditions

Abstract

In the present article, we develop elliptic regularity theory for solutions \(u: \dot \Sigma \to M\) of the following Hamiltonian perturbed contact instanton equation \[ (du - X_H \otimes dt)^{\pi(0,1)} = 0, \quad d(e^{g_{H, u}}u^*(\lambda + H \otimes \gamma)\circ j) = 0 \] under the Legendrian boundary condition associated to a contact triad \((M,\lambda,J)\) and contact Hamiltonian \(H\). We first establish a global \(W^{2,2}\) bound by the Hamiltonian calculus and the harmonic theory of the vector-valued one form \(d_Hu : = du - X_H(u)\otimes \gamma\) and its relevant Weitzenb\"ock formulae utilizing the contact triad connection of the contact triad \((M,\lambda, J)\). Then we establish \(C^{k,\alpha}\)-estimates by an alternating boot-strap argument between the \(\pi\)-component of \(d_Hu\) and the Reeb-component of \(d_Hu\). Based on this regularity theory, we prove an asymptotic \(C^\infty\) convergence result at a puncture under the hypothesis of finite energy. Along the way, we also do the following: (1) We identify the correct choice of the action functional for perturbed contact Hamiltonian trajectories and derive its first variation formula, (2) we identify the correct choice of the energy for the relevant perturbed contact instantons. and (3) we also establish the boundary regularity theorem of \(W^{1,2}\)-weak solutions of perturbed contact instanton equation under the weak Legendrian boundary condition.