K-energy on polarized compactifications of Lie groups

TL;DR

This paper investigates the properness and minimizers of Mabuchi's K-energy on compactifications of reductive Lie groups, providing criteria and alternative proofs for existence of Kähler-Einstein metrics.

Contribution

It offers a new criterion for K-energy properness on polarized compactifications and an alternative proof of Delcroix's theorem for Kähler-Einstein metrics.

Findings

Criterion for K-energy properness on invariant Kähler potentials

Alternative proof of Delcroix's theorem for Fano manifolds

Analysis of minimizers of K-energy in various Kähler classes

Abstract

In this paper, we study Mabuchi's K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K \times K-invariant Kahler potentials. In particular, it turns to give an alternative proof of Delcroix's theorem for the existence of Kahler-Einstein metrics in case of Fano manifolds M . We also study the existence of minimizers of K-energy for general Kahler classes of M.