The classification of Lagrangians nearby the Whitney immersion

Abstract

The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number $+1.$ This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal cubic curve inside the projective plane, and the latter Weinstein manifold is thus the `standard' neighbourhood of Lagrangian immersions of this type. We classify the Lagrangians inside such a neighbourhood which are homologically essential, and which either are embedded or immersed with a single double point; they are shown to be Hamiltonian isotopic to either product tori, Chekanov tori, or rescalings of the Whitney immersion.

72 pages, 16 figures. Revised version: Section 5 has been streamlined

References in corpus (2)

• Lagrangian isotopy of tori in $S^2\times S^2$ and $\mathbb{C}{P}^2$
• Hamiltonian unknottedness of certain monotone Lagrangian tori in $S^2\times S^2$

Cited by in corpus (1)

• On weakly exact Lagrangians in Liouville bi-fillings