Regularity of the geodesic equation in the space of Sasakian metrics

TL;DR

This paper investigates the regularity of geodesics in the space of Sasakian metrics by reducing the problem to a degenerate complex Monge-Ampère equation and establishing a priori estimates for existence and uniqueness.

Contribution

It introduces a reduction of the geodesic equation to a Monge-Ampère type problem and provides new a priori estimates ensuring regularity and uniqueness of geodesics in Sasakian geometry.

Findings

Established existence of $C^{2}_{w}$ geodesics between any two points

Derived a priori estimates for degenerate Monge-Ampère equations

Applied estimates to geometric problems in Sasakian manifolds

Abstract

This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Amp\'ere type eqution on the K\"ahler cone; secondly, we obtain a priori etimates for the above equation. These a priori estimates guarantee the existence and uniqueness of $C^{2}_{w}$ geodesic for any two points in the space of Sasakian metrics. We also give some geometric applications of the above estimates in the end of this paper.