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arXiv · 2108.04337

Local Unknottedness of Planar Lagrangians with Boundary

Zi-Xuan Wang
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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.

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Zi-Xuan Wang. 2021-08-09. Local Unknottedness of Planar Lagrangians with Boundary. https://arxiv.org/abs/2108.04337

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