Kähler-Einstein Metrics

We recall a simple formula for a Kähler-Einstein metric on the unit ball and on the Siegel upper half space, both together with real holomorphic vector fields and consider generalized complex ellipsoids in $C^n$ and show that the logarithm of the defining function, as a potential function, provides a pseudometric, which is Kähler-Einstein. In addition we prove that the complex ellipsoids, endowed with this pseudometric have a real holomorphic vector field, which has several far-reaching differential geometric and functional analytic consequences. Finally we give an example of a real holomorphic vector field of higher order.