Invariants of Lagrangian surfaces

Abstract

We define a nonnegative integer $\la(L,L_0;\phi)$ for a pair of diffeomorphic closed Lagrangian surfaces $L_0,L$ embedded in a symplectic 4-manifold $(M,\w)$ and a diffeomorphism $\phi\in\Diff^+(M)$ satisfying $\phi(L_0)=L$. We prove that if there exists $\phi\in\Diff^+_o(M)$ with $\phi(L_0)=L$ and $\la(L,L_0;\phi)=0$, then $L_0,L$ are symplectomorphic. We also define a second invariant $n(L_1,L_0;[L_t])=n(L_1,L_0,[\phi_t])$ for a smooth isotopy $L_t=\phi_t(L_0)$ between two Lagrangian surfaces $L_0$ and $L_1$ with $\la (L_1,L_0;\phi_1)=0$, which serves as an obstruction of deforming $L_t$ to a Lagrangian isotopy with $L_0,L_1$ preserved.