Toric self-dual Einstein metrics on compact orbifolds (Q2496928)

Let \(X\) be a compact self-dual Einstein 4-orbifold, isometric to a quaternionic Kähler quotient of a quaternionic projective space \({\mathbb H}P^{k-1}\), \(k\geq 2\), by a \((k-2)\)-dimensional subtorus of \(\text{Sp}_k\). In the paper [J. Differ. Geom. 60, No. 3, 485--521 (2002; Zbl 1067.53034)], the first author and \textit{H. Pedersen} proved that any 4-dimensional self-dual Einstein metric, with non-zero scalar curvature and a 2-torus in its isometry group, is locally determined by a real valued function \(F\) on the hyperbolic plane \(\mathcal H\), which is an eigenfunction of the hyperbolic Laplacian with eigenvalue \(3/4\); furthermore, if \(X\) is a quaternionic Kähler quotient as described above, then such eigenfunction is of the form \[ F(\rho, \eta)= \sum_{i = 1}^k \frac{\sqrt{a_i^2 \rho^2 + (a_i \eta - b_i)^2}} {\sqrt{\rho}}\tag{1} \] for some \(a_i, b_i \in \mathbb R\) and where \((\rho>0, \eta)\) are the half-space coordinates of \(\mathcal H\). In this paper, the authors show that any compact self-dual Einstein 4-manifold \(Y\) with positive scalar curvature whose isometry group contains a 2-torus is (up to orbifold coverings) isometric to a quaternionic Kähler quotient \(X\) as defined above. The key point of the proof consists in showing that the function \(F\) associated with the metric on the open set \(W\subset Y\), on which the 2-torus acts freely, has a boundary behaviour which implies that \(F\) is of the form (1). The authors give also a characterization, in terms of the intersection forms, of the compact, simply connected, oriented toric 4-orbifolds admitting a self-dual Einstein metric of positive scalar curvature and with totally geodesic singular orbits of the toric actions.