Geometrical pluripotential theory on Sasaki manifolds

We extend profound results in pluripotential theory on Kahler manifolds to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy. One main result is to generalize T. Darvas theory on the geometric structure of the space of Kahler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kahler structure.