Anti-symplectic involutions for Lagrangian spheres in a symplectic quadric surface

TL;DR

This paper proves that all anti-symplectic involutions with a Lagrangian sphere fixed point set in a monotone S^2×S^2 are connected through Hamiltonian isotopies, revealing a unified structure of such involutions.

Contribution

It establishes the connectedness and Hamiltonian isotopy classification of anti-symplectic involutions fixing a Lagrangian sphere in a monotone symplectic quadric surface.

Findings

The space of these involutions is connected.

Any two such involutions are Hamiltonian isotopic.

The result applies specifically to the monotone S^2×S^2 case.

Abstract

We show that the space of anti-symplectic involutions of a monotone $S^2\times S^2$ whose fixed points set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic.