Toric LeBrun metrics and Joyce metrics

TL;DR

This paper demonstrates that on the connected sum of complex projective planes, toric LeBrun metrics are conformally equivalent to Joyce metrics with a semi-free circle action, using explicit connection forms.

Contribution

It provides an explicit conformal equivalence between toric LeBrun and Joyce metrics on certain complex surfaces, with a novel connection form construction.

Findings

Toric LeBrun metrics can be identified with Joyce metrics via conformal equivalence.

Explicit connection forms for toric LeBrun metrics are constructed.

The equivalence involves a semi-free circle action.

Abstract

We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.