Approximating Ricci solitons and quasi-Einstein metrics on toric surfaces

TL;DR

This paper develops a numerical method to approximate Ricci solitons and quasi-Einstein metrics on toric Kähler surfaces, successfully recovering known solutions and exploring new cases where existence is uncertain.

Contribution

A general numerical approach for prescribed Ricci curvature problems on toric Kähler manifolds, applied to Ricci solitons and quasi-Einstein metrics, including new approximations.

Findings

Recovered Koiso--Cao soliton and Lü--Page--Pope quasi-Einstein metrics.

Numerically approximated Wang--Zhu soliton where explicit form is unknown.

Conducted extensive numerical investigation on quasi-Einstein metrics on P^2P^2P^2 with open existence questions.

Abstract

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