 

Stuart James Hall and Thomas Murphy
Approximating Ricci solitons and quasiEinstein metrics on toric surfaces view print


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


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 quasiEinstein metrics. We begin by recovering the KoisoCao soliton and the LüPagePope quasiEinstein metrics on CP^{2}♯\overline{CP}^{2} (in both cases the metrics are known explicitly). We also find numerical approximations to the WangZhu soliton on CP^{2}♯ 2\overline{CP}^{2} (here the metric is not known explicitly). Finally, a substantial numerical investigation of the quasiEinstein equation on CP^{2}♯ 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 quasiEinstein equation to the same degree of accuracy as the approximations to the WangZhu soliton solve the Ricci soliton equation.


Acknowledgements
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
stuart.hall@mmu.ac.uk
Thomas Murphy:
Department of Mathematics, California State University Fullerton, 800 N. State College Bld., Fullerton, CA 92831, USA.
tmurphy@fullerton.edu

