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

Abstract

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 $M=Q\times Q\times \mathbb{R}$.

v4) 82 pages, application to Shelukhin's conjecture separated away to a separate paper [Ohb], definition of tameness modified, references updated, introduction partially rewritten; v5) 67 pages, format changed to 11 pts, some sections rearranged; v6) 71 pages, errors in the choice of 2-parameter family isotopies and in the definition of $J$-quasi-convexity corrected, over all exposition improved

Cited by in corpus (2)

• The persistence of a relative Rabinowitz-Floer complex
• Contact Hamiltonian dynamics and perturbed contact instantons with Legendrian boundary condition