We will present a simple pluripotential theoretic proof of the uniform estimate for the complex Monge-Ampère equation on compact Kähler manifolds. One of its advantages is that it allows one to use only the local version of Kołodziej's L^p-estimate in order to get the global version. Secondly, the argument is easily adaptable to the case of Hermitian manifolds, thus giving an alternative proof of the uniform estimate recently obtained by Tosatti and Weinkove.

This is joint work with Yanir Rubinstein. We study the initial value problem for the geodesic equation in the space of Kähler metrics in a fixed class. We frame a general conjecture on the solution of the problem and verify it for toric Kähler manifolds and Abelian varieties (with torus invariant metrics). It turns out that except for special initial velocities, the solution only has a finite lifespan in this case.