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.

Preliminary version, all comments are welcome!

References in corpus (3)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• On 4-dimensional gradient shrinking solitons

Cited by in corpus (1)

• Berger curvature decomposition, Weitzenböck formula, and canonical metrics on four-manifolds