Symplectic Rigidity of Fibers in Cotangent Bundles of Open Riemann Surfaces

Abstract

We study symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our main result can be seen as a generalization to open Riemann surfaces of arbitrary genus of work of Eliashberg and Polterovich on the Nearby Lagrangian Conjecture for $T^{\ast }{\mathbb{R}}^2$. As a corollary, we answer a strong version in dimension $2n=4$ of a question of Eliashberg about linking of Lagrangian disks in $T^{\ast }{\mathbb{R}}^n$, which was previously answered by Ekholm and Smith in dimensions $2n\ge 8$.

arXiv comment28 pages, 3 figures. This paper subsumes Section 2 of v1 of arxiv:1909.04753 by the same authors (and goes much further). v2: expanded exposition