Contact instantons with Legendrian boundary condition: a priori estimates, asymptotic convergence and index formula

In Part I, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential $C^\infty$-convergence result at a puncture under the uniform $C^1$ bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which 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. In Part II, we derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [Oh21a]-[Oh22b] and [OYb] containing applications to contact topology and contact Hamiltonian dynamics.