Approximating Ricci solitons and quasi-Einstein metrics on toric surfaces
نویسندگان
چکیده
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 CP2]CP (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang–Zhu soliton on CP2]2CP (here the metric is not known explicitly). Finally, a substantial numerical investigation of the quasi-Einstein equation on CP2]2CP 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.
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