A Computationally Assisted Proof of the Kähler-Einstein Metric on the Third del Pezzo Surface

Abstract

The third del Pezzo surface is known to admit a unique Kähler–Einstein metric, though no closed-form expression for this metric is known. Finding this Einstein metric amounts to solving a complex Monge–Ampère equation, for which only approximate numerical solutions have been obtained. Therefore, while we know such a metric exists, we are unable to make conclusions about relevant geometric quantities, such as the size of the first Laplacian eigenvalue. In this talk, I will present ongoing work towards proving that the true Kähler-Einstein metric is close to a certain numerical approximation, thereby allowing us to make definitive statements about the geometry of this Einstein metric. I will also discuss the prospects of applying these techniques to the study of other four-dimensional Einstein manifolds.