In a Calabi-Yau manifold, the set of all hamiltonian isotopic positive Lagrangian submanifolds can be induced with a structure of a Riemannian manifold. This gives rise to the notion of geodesics between positive Lagrangians. Such geodesics are a solution to a PDE. They have been shown by Solomon-Yuval to be equivalent to a 1-parametric family of imaginary special Lagrangian submanifolds - solutions to a family of elliptic PDEs, via the cylindrical transform.
In this talk we utilize the hyperkähler structure of T^*S^2 to identify imaginary special Lagrangians with J-holomorphic curves. This allows us to construct, under certain assumptions, a geodesic between positive Lagrangians in T^*S^2.