Approximating Ricci solitons and quasi-Einstein metrics on toric surfaces

Abstract

We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kähler 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 Lü--Page--Pope quasi-Einstein metrics on ${\mathbb{CP}}^2♯{\overline{\mathbb{CP}}}^2$ (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang--Zhu soliton on ${\mathbb{CP}}^2♯2{\overline{\mathbb{CP}}}^2$ (here the metric is not known explicitly). Finally, a substantial numerical investigation of the quasi-Einstein equation on ${\mathbb{CP}}^2♯2{\overline{\mathbb{CP}}}^2$ 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.

arXiv comment20 pages, 11 tables