I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion.
Consider $(S^2\times S^2,\omega_{std})$ the product of two spheres with standard product symplectic structure. $L\subset M$ is a Lagrangian torus in M. Given a family $\omega_s$ of forms, such that $\omega_0=\omega_{std}$ Choose $J_s$ an almost complex structure compatible with $\omega_s$. (which we are allowed to choose generically).
Let $u_s$ be a family of Maslov 2 discs with boundary on $L$, converging to some bubbled curve $v$.
Is there something I can say about the shape of the limit curve? The best case scenario would be a statement "If $J_s$ is generic, then the limit consists of two discs, one of Maslov 2 and the other of Maslov 0"
I think I can rule out sphere bubbles by area considerations (A sphere in $S^2\times S^2$ has more area than the original disc), But as for discs I don't know how this works. Can I get in the limit discs with negative Maslov? Discs with Maslov $\ge 4$? a lot of Maslov 0 discs?
Thank you very much
