On the $C^{1,1}$ regularity of geodesics in the space of K\"ahler   metrics
  We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampere equation, which is independent of a positive lower bound for the right hand side. 
Jianchun Chu, Valentino Tosatti, Ben Weinkove
math.DG math.CV