Hi,

I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the zero-section of the cotangent bundle $T^{*}M$, $M$ admits a complexification. By Stenzel and Guillemin this complexification is called the adapted complex structure, its even Kähler. In this complexification the phase function $\varphi$ solves the homogenous Monge-Ampere equation. My question is now: are there Kähler strucutures on a neighbourhood $U$ of the zero-section in the cotangent bundle of $M$, which turn $M$ into a Lagrangian submanifold and such that $M$ is isometrically embedded in $U$, BUT differ from the adapted complex structure (up to biholomorphism)? Do all such kind of structures come from an adapted complex structure? Is this allredy known or can it be derived easilly?