On Lagrangian Tori in $S^2\times S^2$

TL;DR

This paper classifies which toric fibers in a specific symplectic manifold are Hamiltonian isotopic to standard fibers, clarifying the structure of Lagrangian tori in $S^2 imes S^2$.

Contribution

It provides a complete classification of Hamiltonian isotopy classes of toric fibers in the resolved toric degeneration of $S^2 imes S^2$.

Findings

Identifies which toric fibers are Hamiltonian isotopic to standard fibers.

Clarifies the structure of Lagrangian tori in the symplectic manifold.

Completes the classification of toric fibers in this context.

Abstract

In [FOOO12], K. Fukaya, Y. Oh, H. Ohta, and K. Ono (FOOO) obtained the monotone symplectic manifold $S^2\times S^2$ by resolving the singularity of a toric degeneration of a Hirzebruch surface. They identified a continuum of toric fibers in the resolved toric degeneration that are not Hamiltonian isotopic to the toric fibers of the standard toric structure on $S^2\times S^2$. In this paper, we provide a comprehensive classification: for any toric fiber in FOOO's construction of $S^2\times S^2$, we determine whether it is Hamiltonian isotopic to a toric fiber of the standard toric structure of $S^2\times S^2$.