Toric Kahler Metrics: Cohomogeneity One Examples of Constant Scalar Curvature in Action-Angle Coordinates

TL;DR

This paper explores explicit examples of cohomogeneity one constant scalar curvature toric Kähler metrics using action-angle coordinates, highlighting Calabi's family of extremal metrics and their special cases.

Contribution

It provides a detailed analysis of Calabi's U(n)-invariant extremal Kähler metrics within the framework of symplectic action-angle coordinates, illustrating their role as cohomogeneity one constant scalar curvature examples.

Findings

Calabi's family includes many cohomogeneity one constant scalar curvature metrics.

Explicit action-angle coordinate descriptions of these metrics are provided.

The work connects extremal Kähler metrics with concrete geometric examples.

Abstract

In these notes, after an introduction to toric Kahler geometry, we present Calabi's family of U(n)-invariant extremal Kahler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.