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Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry
Mathematical Research Letters ( IF 0.6 ) Pub Date : 2021-03-01
Claude LeBrun

Peng Wu [P. Wu, “Einstein four-manifolds with self-dual Weyl tensor of nonnegative determinant”, Int. Math. Res. Not. (2021), no. 2, 1043–1054] recently announced a beautiful characterization of conformally Kähler, Einstein metrics of positive scalar curvature on compact oriented $4$-manifolds via the condition $\operatorname{det}(W^{+}) \gt 0$. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences of our method, which builds on techniques previously developed in [C. LeBrun, “Einstein metrics, harmonic forms, and symplectic four-manifolds,” Ann. Global Anal. Geom. 48 (2015), no. 1, 75–85].

中文翻译:

爱因斯坦流形,自双Weyl曲率和保形Kähler几何

彭武[P. Wu,“具有非负行列式自对偶Weyl张量的爱因斯坦四流形”,《诠释学》。数学。Res。不是。(2021年),没有。[2,1043–1054]最近宣布了通过条件$ \ operatorname {det}(W ^ {+})\ gt 0 $对紧致的$ 4 $流形上的正标量曲率的爱因斯坦共形Kähler,Einstein度量的优美刻画。在本说明中,我们通过提供与他的结果完全不同的证据来支持他的主张。然后,我们将基于先前在[C. LeBrun,“爱因斯坦度量,调和形式和辛四流形”,安。全球肛门。几何 48(2015),第。1,75–85]。
更新日期:2021-04-19
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