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 knwonknown fact that in a neighbourhood $U$ of the zero-section section of the cotangent bundle $T^{*}M$, $M$ admits a complexification. By Stenzel and Guillemin this complexification is called the adapted complex structure, its. It is even Kähler. In this complexification, the phase function $\varphi$ solves the homogenous Monge-Ampere equation. My question is now: are there Kähler strucuturesstructures on a neighbourhood $U$ of the zero-section 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 allredyalready known or can it be derived easillyeasily?
Dmitri