New York Journal of Mathematics
Volume 22 (2016) 615-635


Stuart James Hall and Thomas Murphy

Approximating Ricci solitons and quasi-Einstein metrics on toric surfaces

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Published: July 14, 2016
Keywords: Ricci soliton, Einstein metric, quasi-Einstein metric, toric Kähler geometry, numerical approximation
Subject: Primary 53C25, 53C55; secondary 65D99

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♯\overline{CP}2 (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang-Zhu soliton on CP2♯ 2\overline{CP}2 (here the metric is not known explicitly). Finally, a substantial numerical investigation of the quasi-Einstein equation on CP2♯ 2\overline{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.


A good deal of the work was conducted whilst SH was visiting TM at California State University Fullerton. SH would like to thank the department for their hospitality. This trip was funded by a CSUF Startup grant.

Author information

Stuart James Hall:
School of Computing, Mathematics and Digital Technology, John Dalton Building, Manchester Metropolitan University, Chester Street, Manchester, M1 5GD

Thomas Murphy:
Department of Mathematics, California State University Fullerton, 800 N. State College Bld., Fullerton, CA 92831, USA.