A Weitzenböck formula for canonical metrics on four-manifolds

Wu, Peng

doi:10.1090/tran/6964

A Weitzenböck formula for canonical metrics on four-manifolds
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Trans. Amer. Math. Soc. 369 (2017), 1079-1096 Request permission

Abstract:

We first provide an alternative proof of the classical Weitzenböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds. As applications, we classify Einstein four-manifolds and conformally Einstein four-manifolds with half two-nonnegative curvature operator, which in some sense provides a characterization of Kähler-Einstein metrics and Hermitian, Einstein metrics with positive scalar curvature on four-manifolds, respectively. We also discuss the classification of four-dimensional gradient shrinking Ricci solitons with half two-nonnegative curvature operator and half harmonic Weyl curvature.

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