Which symplectic forms on S2xS2 have Lagrangian Klein bottles?

Abstract

Using (almost) toric fibrations and their visible Lagrangians we can construct many novel and interesting examples of Lagrangian submanifolds of symplectic 4 manifolds. Naturally, one can ask whether visible Lagrangians are all the Lagrangians that exist, or, in other words, how faithful the pictures coming from almost toric fibrations are. I will answer this question for Klein bottles in (S2xS2,omega_lambda), i.e. the product of two spheres where the first factor has area 1 and the other factor has area lambda. In particular, I will first construct a visible Lagrangian Klein bottle when lambda<2. Then I will show that no Lagrangian Klein bottles exist otherwise. The key input for obstructing the existence of the Klein bottles is Luttinger surgery along with techniques of (compact) pseudoholomorpic curves and Seiberg-Witten theory. This is joint work with J. Evans.