Geometrical pluripotential theory on Sasaki manifolds

TL;DR

This paper extends key pluripotential theory results from Kahler manifolds to Sasaki manifolds, facilitating the study of constant scalar curvature Sasaki metrics through a generalized geometric framework.

Contribution

It generalizes Darvas' theory on the geometry of Kähler potentials to the Sasaki setting, expanding pluripotential theory tools to this broader context.

Findings

Generalization of Darvas' theory to Sasaki manifolds

Extension of pluripotential theory results to Sasaki setting

Foundation for solving Sasaki cscs metric existence problem

Abstract

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.