Self-dual metrics on toric 4-manifolds: extending the Joyce construction

Abstract

Toric geometry studies manifolds M2n acted on effectively by a torus of half their dimension, Tn. Joyce shows that for such a 4-manifold sufficient conditions for a conformal class of metrics on the free part of the action to be self-dual can be given by a pair of linear ODEs and gives criteria for a metric in this class to extend to the degenerate orbits. Joyce and Calderbank-Pedersen use this result to find representatives which are scalar flat K¨ahler and self-dual Einstein respectively. We review some results concerning the topology of toric manifolds and the construction of Joyce metrics. We then extend this construction to give explicit complete scalar-flat K¨ahler and self-dual Einstein metrics on manifolds of infinite topological type, and to find a new family of Joyce metrics on open submanifolds of toric spaces. We then give two applications of these extensions — first, to give a large family of scalar flat K¨ahler perturbations of the Ooguri-Vafa metric, and second to search for a toric scalar flat K¨ahler metric on a neighbourhood of the origin in C2 whose restriction to an annulus on the degenerate hyperboloid {(z1, z2)|z1z2 = 0} is the cusp metric.