Numerical approximations to extremal toric K\"ahler metrics with arbitrary K\"ahler class

TL;DR

This paper introduces new algorithms for approximating extremal toric K"ahler metrics, demonstrating their effectiveness on complex surfaces and providing insights into the stability of related Einstein metrics.

Contribution

The paper develops novel algorithms for approximating extremal toric K"ahler metrics and applies them to a specific complex surface, comparing results with existing methods.

Findings

Approximate extremal toric K"ahler metrics effectively.

Numerical evidence of conformal instability of the Einstein metric.

Comparison with previous approximation methods.

Abstract

We develop new algorithms for approximating extremal toric K\"ahler metrics. We focus on an extremal metric on $\mathbb{CP}^{2}\sharp2\overline{\mathbb{CP}}^{2}$, 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.