Lagrangian isotopies in Stein manifolds

Studying the space of Lagrangian submanifolds is a fundamental problem in symplectic topology. Lagrangian spheres appear naturally in the Leftschetz pencil picture of symplectic manifolds. In this paper we demonstrate the uniqueness up to Hamiltonian isotopy of the Lagrangian spheres in some 4-dimensional Stein symplectic manifolds. The most important example is the cotangent bundle of the 2-sphere, T S, with its standard symplectic structure. We recall that if a convex symplectic manifold has a boundary of contacttype, then we can perform surgery operations on the manifold by adding handles to the boundary. In the 4-dimensional case these handles can be of index 1 or 2. Our other examples are symplectic manifolds formed by adding 1-handles to a unit cotangent bundle T S. Questions regarding Lagrangian isotopy classes are independent of which metric we use to define a unit tangent bundle or of any choices involved in adding 1-handles.