The Ding functional, Berndtsson convexity and moment maps

TL;DR

This paper explores the connection between the Ding functional, Berndtsson convexity, and moment maps in the context of Kähler-Einstein metrics, providing new insights into their geometric structure and applications to the Kähler-Ricci flow.

Contribution

It develops a framework linking the Ding functional and Berndtsson convexity via moment maps, offering a novel geometric perspective on Kähler-Einstein metrics.

Findings

Interpretation of Berndtsson convexity as a metric on complex structures

Application of moment map framework to Kähler-Ricci flow

Enhanced understanding of the geometric structure of Kähler-Einstein metrics

Abstract

We explain how the formal aspects of the theory of Kahler-Einstein metrics can be developed in the framework of moment maps. The central result we use is the Berndtsson convexity theorem, which is interpreted as defining a metric on the space of complex structures. We discuss some applications of these ideas to the Kahler-Ricci flow.