I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. This complex structure $J$ is unique due to the Bruhat and Whitney complexification method. With this complex structure there comes a symplectic form $\omega$ that is Kähler and moreover $M$ is a Lagrangian submanifold in this complexified neighbourhood of the zero section in the cotangent bundle. My question is: is this Kähler form $\omega$ unique? Does there exists a different (or in the same cohomology) Kähler form with the same properties? If its not unique to which extend does uniqueness fail? Or, how can one characterize the space of Kähler forms on a neighbourhood of the zero section in the cotangent bundle that fixes $M$ as a Lagrangian submanifold? (Actually I am very interested in the last question.)