Regularity of the geodesic equation in the space of Sasakian metrics

In this paper, we study a geodesic equation in the space of Sasakian metrics H. The equation leads to the Dirichlet problem of a complex Monge–Ampère type equation on the Kähler cone. This equation differs from the standard complex Monge–Ampère equation in a significant way, with gradient terms involved in the (1,1) symmetric tensor of the operator. We establish appropriate regularity estimates for this complex Monge–Ampère type equation. As geometric application, we show that the space of Sasakian metrics H is a metric space, and the constant transversal scalar curvature metric realizes the global minimum of K-energy if the first basic Chern class . We also prove that the constant transversal scalar curvature metric is unique in each basic Kähler class if the first basic Chern class is either strictly negative or zero.