Approximating Ricci solitons and quasi-Einstein metrics on toric surfaces

We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kahler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso-Cao soliton and the Lu-Page-Pope quasi-Einstein metrics on CP2#(CP) over bar (2) (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang-Zhu soliton on CP2#2 (CP) over bar (2) (here the metric is not known explicitly). Finally, a substantial numerical investigation of the quasi-Einstein equation on CP2#2 (CP) over bar (2) is conducted. In this case it is an open problem as to whether such metrics exist on this manifold. We find metrics that solve the quasi-Einstein equation to the same degree of accuracy as the approximations to the Wang-Zhu soliton solve the Ricci soliton equation.