Existence of higher extremal Kahler metrics on a minimal ruled surface

In this paper we prove that on a special type of minimal ruled surface, which is an example of a 'pseudo-Hirzebruch surface', every Kahler class admits a certain kind of 'higher extremal Kahler metric', which is a Kahler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kahler metric. From an already proven result, it will follow that this specific higher extremal Kahler metric cannot be a 'higher constant scalar curvature Kahler (hcscK) metric', which is defined, again by analogy with the definition of a constant scalar curvature Kahler (cscK) metric, to be a Kahler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any Kahler class on this surface.(c) 2023 Elsevier Masson SAS. All rights reserved.