Contact instantons with Legendrian boundary condition: a priori estimates, asymptotic convergence and index formula. (English) Zbl 1543.53081
Summary: In this paper, 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. Leaving the study of \(C^1\)-estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [the first author, “Geometry and analysis of contact instantons and entanglement of Legendrian links I”, Preprint, arXiv:2111.02597], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [the first author, loc. cit.; “Contact instantons, anti-contact involution and proof of Shelukhin’s conjecture”, Preprint, arXiv:2212.03557; “Gluing theories of contact instantons and of pseudoholomorphic curves in SFT”, Preprint, arXiv:2205.00370] and [the authors, Int. J. Math. 35, No. 7, Article ID 2450019, 60 p. (2024; Zbl 07863346)] containing applications to contact topology and contact Hamiltonian dynamics.
MSC:
| 53D42 | Symplectic field theory; contact homology |
| 53D35 | Global theory of symplectic and contact manifolds |
| 58J32 | Boundary value problems on manifolds |
| 32Q65 | Pseudoholomorphic curves |
Keywords:
contact manifolds; Legendrian submanifolds; contact instantons; iso-speed Reeb chords; asymptotic contact charge; contact triad connections; graded bridged Legendrian links; Maslov indexReferences:
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