The formalism to complexify time in the flow of a nonholomorphic vector field on a complex manifold is reviewed. The complexified flow, besides acting on $M$, changes also the complex structure. We will describe the following applications:

1. For a compact Kahler manifold the imaginary time Hamiltonian flows correspond to Mabuchi geodesics in the infinite dimensional space of Kahler metrics on $M$. These geodesics play a very important role in the study of stability of Kahler manifolds. A nontrivial nontoric example on the two-dimensional sphere will be described.
2. Let the compact connected Lie group $G$ act in an Hamiltonian and Kahler way on a Kahler manifold $M$ and assume that its action extends to $G_C$. Then, by taking geodesics of Kahler structures generated by convex functions of the $G$-momentum to infinite geodesic time, one gets (conjecturally always, proved on several important examples) a concentration of holomorphic sections of holomorphic line bundles on inverse images of coadjoint orbits under the $G$-momentum map. A nontrivial toric example and the case of $M=G_C$ will be described.

On work with T Baier, J Hilgert, O Kaya, JP Nunes, M Pereira, P Silva.