On Calabi's diastasis function of the cigar metric

TL;DR

This paper demonstrates that the Cigar metric on the complex plane is a real analytic Kähler manifold with a globally positive Calabi's diastasis function that cannot be Kähler immersed into any complex space form.

Contribution

It provides a counterexample showing the limitations of Kähler immersions for certain metrics with positive diastasis functions.

Findings

Cigar metric has a globally defined positive Calabi's diastasis function.

The Cigar metric cannot be Kähler immersed into any complex space form.

The example challenges assumptions about Kähler immersions of metrics with positive diastasis functions.

Abstract

We show that the Cigar metric on $\mathbb{C}$ is an example of real analytic K\"ahler manifold with globally defined and positive Calabi's diastasis function which cannot be K\"ahler immersed into any (finite or infinite dimensional) complex space form.