Geometry and analysis of contact instantons and entanglement of Legendrian links I

The purposes of the present paper are two-fold. Firstly we further develop the interplay between the contact Hamiltonian geometry and the geometric analysis of Hamiltonian-perturbed contact instantons with the Legendrian boundary condition, which is initiated by the present author in \cite{oh:contacton-Legendrian-bdy}. We introduce the class of \emph{tame contact manifolds} $(M,\lambda)$, which includes compact ones but not necessarily compact, and establish uniform a priori $C^0$-estimates for the contact instantons. Then we study the problem of estimating the Reeb-untangling energy of one Legendrian submanifold from another, and formulate a particularly designed parameterized moduli space for the study of the problem. We establish the Gromov-Floer-Hofer type convergence result for contact instantons of finite energy and construct its compactification of the moduli space, first by defining the correct energy and then by proving uniform a priori energy bounds in terms of the oscillation of the relevant contact Hamiltonian. Secondly, as an application of this geometry and analysis of contact instantons, we prove that the \emph{self Reeb-untangling energy} of a compact Legendrian submanifold $R$ in any tame contact manifold $(M,\lambda)$ is greater than that of the period gap $T_\lambda(M,R)$ of the Reeb chords of $R$. This is an optimal result in general. In a sequel \cite{oh:shelukhin-conjecture}, we also prove Shelukhin's conjecture specializing to the Legendrianization of contactomorphisms of closedcoorientable contact manifold $(Q,\xi)$ and utilizing its $\mathbb Z_2$-symmetry as the fixed point set of anti-contact involution to overcome the \emph{nontameness} of contact product $M = Q \times Q \times \mathbb R$.