Geodesics in the space of Kahler metrics and volume forms

We discuss optimal regularity of geodesics in the space of K\"ahler metrics of a compact K\"ahler manifold, as well as the space of volume forms on a compact Riemannian manifold. They are solutions of nonlinear degenerate elliptic equations: homogeneous complex Monge-Amp\`ere equation and Nahm's equation (introduced by Donaldson), respectively. The highest regularity one can expect is $C^{1,1}$.Non UBCUnreviewedAuthor affiliation: Jagiellonian UniversityFacult