Abstract
Commun.Math.Phys.282:357-393,2008 The third del Pezzo surface admits a unique Kaehler-Einstein metric, which is
not known in closed form. The manifold's toric structure reduces the Einstein
equation to a single Monge-Ampere equation in two real dimensions. We
numerically solve this nonlinear PDE using three different algorithms, and
describe the resulting metric. The first two algorithms involve simulation of
Ricci flow, in complex and symplectic coordinates respectively. The third
algorithm involves turning the PDE into an optimization problem on a certain
space of metrics, which are symplectic analogues of the "algebraic" metrics
used in numerical work on Calabi-Yau manifolds. Our algorithms should be
applicable to general toric manifolds. Using our metric, we compute various
geometric quantities of interest, including Laplacian eigenvalues and a
harmonic (1,1)-form. The metric and (1,1)-form can be used to construct a
Klebanov-Tseytlin-like supergravity solution.