Toric gravitational instantons with four fixed points

We classify the simply connected toric gravitational instantons whose torus action has four fixed points. They are the hyperkähler multi-Eguchi–Hanson and multi-Taub–NUT metrics with four collinear centres, the asymptotically flat instantons on Li and Sun's manifold $X_3$, up to scale one for each twist at infinity in $(0,1)$ for a fixed orientation of the torus, and, up to scale, a single further instanton with an ALF end, which is not hyperkähler. The last two are Hermitian for neither orientation. In particular, the toric asymptotically flat instanton on $X_3$ with a given twist is unique up to scale, as conjectured by Li and Sun. The proof combines the solitonic form of Geroch's conjecture, which turns smoothness into polynomial equations for the soliton data, with a convexity principle adapted from Aksteiner. At a fixed asymptotic parameter, the cone errors of the conical metrics with a given rod structure are the gradient of a strictly convex function of the rod lengths, and the strict convexity gives uniqueness. Existence follows from an explicit solution for the ALF end and, for the AF ends, from Li and Sun's existence theorem. We also outline how a chart of Aksteiner identifies these AF instantons with those found by Teo.