Calabi Energies of Extremal Toric Surfaces

TL;DR

This paper derives a formula linking the scalar curvature norm of extremal Kähler metrics on toric manifolds to the geometry of their moment polytopes, aiding in constructing Bach-flat metrics in complex dimension 2.

Contribution

It provides a new formula for the L^2 norm of scalar curvature purely based on the moment polytope geometry, with applications to 4-manifold metrics.

Findings

Formula for scalar curvature norm in terms of polytope geometry

Application to constructing Bach-flat metrics in 4-manifolds

Enhanced understanding of extremal toric Kähler metrics

Abstract

We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains to the case of complex dimension 2, where it plays a key role in construction of Bach-flat metrics on appropriate 4-manifolds.