On Optimal 4-Dimensional Metrics

TL;DR

This paper classifies simply connected compact oriented 4-manifolds that support scalar-flat, anti-self-dual metrics, including a new proof that the connected sum of five reverse-oriented complex projective planes admits such metrics.

Contribution

It provides a complete classification of these 4-manifolds and proves for the first time that the connected sum of five reverse-oriented complex projective planes admits scalar-flat, anti-self-dual metrics.

Findings

Complete classification of manifolds with these metrics

Proof that the connected sum of five reverse-oriented complex projective planes admits such metrics

Advancement in understanding 4-manifold geometry

Abstract

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented complex projective planes admits such metrics.