A note on higher extremal metrics

TL;DR

This paper introduces higher extremal Kähler metrics, provides examples on minimal ruled surfaces, and proves a perturbation result showing the existence of non-trivial higher constant scalar curvature metrics with harmonic top Chern form.

Contribution

It introduces higher extremal Kähler metrics, constructs examples, and proves a perturbation result for higher constant scalar curvature metrics, advancing understanding of harmonic Chern forms.

Findings

Existence of higher extremal Kähler metrics on minimal ruled surfaces

Perturbation results for higher constant scalar curvature metrics

Short proof of Liu's formula for Bando-Futaki invariants

Abstract

In this paper we introduce higher extremal Kahler metrics. We provide an example of the same on a minimal ruled surface. We also prove a perturbation result that implies that there are non-trivial examples of higher constant scalar curvature metrics, which are basically metrics where the top Chern form is harmonic. We also give a relatively short proof of a formula due to Liu for the Bando-Futaki invariants (which are obstructions for the existence of harmonic Chern forms) of hypersurfaces in projective space.