Numerical approximations to extremal toric Kähler metrics with arbitrary Kähler class

Abstract

We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on ${\mathbb{CP}}^2♯2{\overline{\mathbb{CP}}}^2$, which is conformal to an Einstein metric (the Chen-LeBrun-Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric which gives a numerical evidence that the Einstein metric is conformally unstable under the Ricci flow.

arXiv comment16 pages v2 changes reflecting referees' comments. To appear in Proc. Edin. Math. Soc