Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Abstract

We prove that a four-dimensional gradient shrinking Ricci soliton with $\delta W^{\pm }=0$ is either Einstein, or a finite quotient of $S^3\times \mathbb{R}$, $S^2\times {\mathbb{R}}^2$ or ${\mathbb{R}}^4$. We also prove that a four-dimensional cscK gradient Ricci soliton is either Kähler-Einstein, or a finite quotient of $M\times \mathbb{C}$, where $M$ is a Riemann surface. The main arguments are curvature decompositions, the Weitzenböck formula for half Weyl curvature, and the maximum principle.

arXiv commentPreliminary version, all comments are welcome!

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Cited by in corpus (5)

• Classification of expanding and steady Ricci solitons with integral curvature decay
• On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature
• A gap theorem of four-dimensional gradient shrinking solitons
• Berger curvature decomposition, Weitzenböck formula, and canonical metrics on four-manifolds
• Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations