Toric LeBrun metrics and Joyce metrics (Q372690)

First, the authors present the construction of LeBrun's self dual hyperbolic monopole metrics. They consider a subclass of LeBrun metrics, toric LeBrun metrics (a LeBrun metric admits a torus action if and only if the monopole points belong to a common hyperbolic geodesic). The second class of metrics considered in this paper are the metrics on \(n\mathbb CP^2\) introduced by \textit{D. D. Joyce} in [Duke Math. J. 77, No. 3, 519--552 (1995; Zbl 0855.57028)].NEWLINENEWLINENEWLINEIn the paper an explicit connection for any toric LeBrun metric is found and it is used to prove the main result,NEWLINENEWLINENEWLINETheorem 1.1: On \(n\mathbb CP^2\), the class of toric LeBurn metrics and the class of Joyce metrics admitting a semi-free circle action are the same, and any metric of the first class can be identified with a metric of the second class through an explicit conformal equivalence.