# Kähler-Einstein Metrics

## TL;DR

This paper explores Kähler-Einstein metrics on specific complex spaces and their geometric properties.

## Contribution

The paper introduces a new pseudometric for complex ellipsoids and shows it is Kähler-Einstein.

## Key findings

- The logarithm of the defining function serves as a Kähler-Einstein potential on complex ellipsoids.
- Complex ellipsoids with this pseudometric admit real holomorphic vector fields.
- An example of a higher-order real holomorphic vector field is provided.

## Abstract

We recall a simple formula for a Kähler-Einstein metric on the unit ball and on the Siegel upper half space, both together with real holomorphic vector fields and consider generalized complex ellipsoids in \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {C}^n$$\end{document}Cn and show that the logarithm of the defining function, as a potential function, provides a pseudometric, which is Kähler-Einstein. In addition we prove that the complex ellipsoids, endowed with this pseudometric have a real holomorphic vector field, which has several far-reaching differential geometric and functional analytic consequences. Finally we give an example of a real holomorphic vector field of higher order.