The Dirichlet and the weighted metrics for the space of Kahler metrics

TL;DR

This paper explores the intrinsic geometry of the space of Kähler metrics using Dirichlet and weighted metrics, analyzing their curvature and relationships with other geometric structures.

Contribution

It introduces and studies the Dirichlet and weighted metrics on the space of Kähler metrics, detailing their geometric properties and connections to existing metrics.

Findings

Computed curvature of the Dirichlet metric

Linked the Dirichlet metric to the Calabi metric and K-energy

Investigated properties of weighted metrics

Abstract

In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi metric, the K-energy, the degenerate complex Hessian equation. The second part is on the weighted metrics, for which we investigate as well their geometric properties.