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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Weitzenböck formula for canonical metrics on four-manifolds

Author: Peng Wu
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 53C25; Secondary 53C24
Published electronically: July 26, 2016
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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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Additional Information

Peng Wu
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Address at time of publication: Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China

Keywords: Weitzenb\"ock formula, Berger curvature decomposition, half two-positive curvature operator, half positive isotropic curvature, gradient Ricci soliton, quasi-Einstein manifold, conformally Einstein manifold, generalized quasi-Einstein manifolds, canonical metric, smooth metric measure space.
Received by editor(s): February 9, 2015
Published electronically: July 26, 2016
Article copyright: © Copyright 2016 American Mathematical Society