Local Unknottedness of Planar Lagrangians with Boundary

TL;DR

This paper proves the smooth and Hamiltonian isotopy conjectures for certain 2D Lagrangian submanifolds in cotangent bundles, specifically for the pair of pants and the cylinder, using pseudo-holomorphic curve theory and isotopy constructions.

Contribution

It establishes the smooth nearby Lagrangian conjecture for the pair of pants and the Hamiltonian version for the cylinder, extending previous results with new isotopy techniques.

Findings

Smooth isotopy for the pair of pants Lagrangian

Hamiltonian isotopy for Lagrangian cylinders

Extension of results using pseudo-holomorphic curves

Abstract

We show the smooth version of the nearby Lagrangian conjecture for the 2-dimensional pair of pants and the Hamiltonian version for the cylinder. In other words, for any closed exact Lagrangian submanifold of $T^{*}M$, there is a smooth or Hamiltonian isotopy, when $M$ is a pair of pants or a cylinder respectively, from it to the 0-section. For the cylinder we modify a result of G. Dimitroglou Rizell for certain Lagrangian tori to show that it gives the Hamiltonian isotopy for a Lagrangian cylinder. For the pair of pants, we first study some results from pseudo-holomorphic curve theory and the planar Lagrangian in $T^{*}\mathbb{R}^{2}$, then finally using a parameter construction to obtain a smooth isotopy for the pair of pants.