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Kähler-Einstein Metrics
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DOI：10.1007/s12220-026-02401-4.png)
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Abstract
En 中文
We recall a simple formula for a K & auml;hler-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 Cn\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} and show that the logarithm of the defining function, as a potential function, provides a pseudometric, which is K & auml;hler-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.
Keywords:
K & auml
hler-Einstein metrics
Real holomorphic vector fields