Pinched self-dual Weyl curvature on Einstein four-manifolds
Inyoung Kim
TL;DR
The paper investigates compact oriented Einstein 4-manifolds with harmonic self-dual Weyl curvature and pinching of the self-dual Weyl spectrum. It develops a framework connecting Weyl geometry, almost-Kähler structures, and holomorphic sectional curvature, deriving sharp relations between $H_{omax},H_{omin}$, $s$, $s^{*}$, and the $W^{-}$-spectrum, and showing that pinching plus nonpositive type forces anti-self-duality or a Kähler-like structure on a large open set. By combining Weitzenböck-type arguments with line-bundle considerations and existing conformal classification results (LeBrun, Guan, Polombo, Derdziński), it sharpens pinching criteria and yields partial classifications in the Einstein and Kähler–Einstein cases. The findings contribute to a refined understanding of when curvature pinching forces self-duality, and they connect to broader themes in the Goldberg conjecture and the geometry of constant holomorphic sectional curvature manifolds.
Abstract
We show that a compact oriented riemannian four-manifold with harmonic and pinched self-dual Weyl curvature is anti-self-dual if the type is nonpositive. The main part is to show that there is an almost-Kähler structure outside the zero set of the self-dual Weyl curvature.