Monge-Amp\`ere operators, energy functionals, and uniqueness of Sasaki-extremal metrics

Understand

We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold.

• We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds.
• This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics minimizing it.
• More generally, a relative version of the K-energy is convex, and bounded below if there exists a Sasaki-extremal metric, providing an important necessary condition for Sasaki-extremal metrics.