르장드르 경계 조건을 갖춘 접촉 순간자

Abstract

This dissertation combines three pivotal papers [OY24, OYa, OYb] on contact instantons with Legendrian boundary condition and generating functions in contact geometry. The research advances the analytic and topological foundations necessary for the study of Legendrian Floer theory and provides new tools for exploring the structures of Legendrian submanifolds.

We begin by establishing 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∞- convergence result at a puncture under the uniform C1 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. We also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the subsequent sequels containing applications to contact topology and contact Hamiltonian dynamics.

Next, we formulate a contact analogue on the one-jet bundle J1B of Weinstein’s observation which reads the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of the Legendrian generating function which is the Legendrian counterpart of Laudenbach-Sikorav’s canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which utilizes a finite dimensional approximation of the action functional.

Finally, we develop the Floer-style elliptic Morse theory for the Hamiltonian-perturbed contact action functional, motivated by the construction of canonical generating function, and assoicate the Legendrian contact instanton cohomology, denote by HI∗(J1B;H, J), to each Legendrian submanifold ψH(oJ1B) contact isotopic to the zero section of one-jet bundle. Then we give a Floer theoretic construction of Legendrian spectral invariants and establish their basic properties. This work applies the earlier analysis to the one-jet bundle case. This theory subsumes the Lagrangian intersection theory and spectral invariants on the cotangent bundle previously developed in [Oh97, Oh99], and its extension to exact immersed Lagrangian submanifolds. The main ingredient for the study is the interplay between the geometric analysis of the Hamiltonian-perturbed contact instantons and the calculus of contact Hamiltonian geometry.