Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition

This is the first of a series of papers in preparation in which we study the Hamiltonian perturbed contact instantons with Legendrian boundary condition and its applications. In this paper, we prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes, which eliminates the phenomenon of the appearance of `spiraling cusp instanton along a Reeb core'. This removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. We also extend the a priori elliptic coercive estimates for the contact instantons with boundary, and prove an asymptotic exponential $C^\infty$-convergence result at a puncture under the uniform $C^1$ bound. In a sequel to the present paper, we study the $C^1$ estimates by defining a proper notion of energy for the contact instantons, and develop a Fredholm theory and construct a Gromov-type compactification of the moduli space of contact instantons with Legendrian boundary condition and of finite energy, and apply them to problems in contact topology and dynamics.