2021/11/03 by Yong‐Geun Oh, Oh, Yong-Geun
Mathematics · #53D42 #58J32 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2111.02597
openalex publication_date 2021/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purposes of the present paper are two-fold. Firstly we further develop\nthe interplay between the contact Hamiltonian geometry and the geometric\nanalysis of Hamiltonian-perturbed contact instantons with the Legendrian\nboundary condition, which is initiated by the present author in\n citeoh:contacton-Legendrian-bdy. We introduce the class of \tame\ncontact manifolds (M,\λ), which includes compact ones but not\nnecessarily compact, and establish uniform a priori C0-estimates for the\ncontact instantons. Then we study the problem of estimating the Reeb-untangling\nenergy of one Legendrian submanifold from another, and formulate a particularly\ndesigned parameterized moduli space for the study of the problem. We establish\nthe Gromov-Floer-Hofer type convergence result for contact instantons of finite\nenergy and construct its compactification of the moduli space, first by\ndefining the correct energy and then by proving uniform a priori energy bounds\nin terms of the oscillation of the relevant contact Hamiltonian. Secondly, as\nan application of this geometry and analysis of contact instantons, we prove\nthat the \self Reeb-untangling energy of a compact Legendrian submanifold\nR in any tame contact manifold (M,\λ) is greater than that of the\nperiod gap T_\λ(M,R) of the Reeb chords of R. This is an optimal\nresult in general. In a sequel citeoh:shelukhin-conjecture, we also prove\nShelukhin's conjecture specializing to the Legendrianization of\ncontactomorphisms of closedcoorientable contact manifold (Q,\ξ) and\nutilizing its mathbb Z2-symmetry as the fixed point set of anti-contact\ninvolution to overcome the \nontameness of contact product M = Q \×\nQ \× mathbb R.\n