Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition

TL;DR

This paper establishes foundational results for contact instantons with Legendrian boundary conditions, proving vanishing asymptotic charge and exponential convergence, thus advancing the compactification and Fredholm theory essential for contact topology applications.

Contribution

It proves the vanishing of asymptotic charge for contact instantons with Legendrian boundary, enabling progress in their compactification and Fredholm theory.

Findings

Asymptotic charge at punctures vanishes under Legendrian boundary condition.

Exponential $C^ abla$-convergence at punctures with uniform $C^1$ bounds.

Extension of elliptic coercive estimates for contact instantons with boundary.

Abstract

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.