Four-manifolds of Pinched Sectional Curvature

TL;DR

This paper investigates four-dimensional closed manifolds under new pinching curvature conditions, establishing their definiteness, self-duality, or specific Einstein metrics, and classifies Einstein manifolds with positive intersection form and bounded sectional curvature.

Contribution

It introduces novel pinching curvature conditions that lead to classification results for four-manifolds, including definiteness, self-duality, and specific Einstein metrics.

Findings

Manifolds are definite under certain pinching conditions.

Harmonic Weyl tensor implies self-duality or anti-self-duality.

Classifies Einstein manifolds with positive intersection form and curvature bounds.

Abstract

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere or their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature.