The classification of Lagrangians nearby the Whitney immersion

TL;DR

This paper classifies certain Lagrangian submanifolds near the Whitney immersion in a four-dimensional symplectic space, identifying them as Hamiltonian isotopic to known models like product tori, Chekanov tori, or rescaled Whitney immersions.

Contribution

It provides a classification of homologically essential Lagrangians near the Whitney immersion, including embedded and immersed cases, expanding understanding of their Hamiltonian isotopy classes.

Findings

Lagrangians are Hamiltonian isotopic to product tori, Chekanov tori, or rescaled Whitney immersions.

Classification applies to both embedded and immersed Lagrangians with a single double point.

Results connect the Whitney immersion to standard Lagrangian models in symplectic topology.

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.