Berger curvature decomposition, Weitzenb\"ock formula, and canonical metrics on four-manifolds

TL;DR

This paper offers a new proof of the Weitzenböck formula for Einstein four-manifolds using Berger curvature decomposition and develops a unified framework for canonical metrics, leading to classification results for specific curvature conditions.

Contribution

It introduces an alternative proof of the Weitzenböck formula via Berger decomposition and establishes a unified approach for analyzing canonical metrics on four-manifolds.

Findings

Classification of Einstein four-manifolds with half two-nonnegative curvature operator

Characterization of Kähler-Einstein metrics with positive scalar curvature

Discussion of four-manifolds with half harmonic Weyl curvature

Abstract

We first provide an alternative proof of the classical Weitzneb\"ock formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenb\"ock formula for a large class of canonical metrics on four-manifolds (or a Weitzenb\"ock formula for "Einstein metrics" on four-dimensional smooth metric measure spaces). As applications, we classify Einstein four-manifolds of half two-nonnegative curvature operator, which in some sense provides a characterization of K\"ahler-Einstein metrics with positive scalar curvature on four-manifolds, we also discuss four-manifolds of half two-nonnegative curvature operator and half harmonic Weyl curvature.