Four-dimensional gradient shrinking solitons with pinched curvature

Zhang, Zhu-Hong

doi:10.1090/proc/13859

Four-dimensional gradient shrinking solitons with pinched curvature
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Proc. Amer. Math. Soc. 146 (2018), 3049-3056 Request permission

Abstract:

We show that any four-dimensional gradient shrinking soliton with pinched Weyl curvature $(*)$ and satisfying $c_1 \le R \le c_2$ for some positive constant $c_1$ and $c_2$, will have nonnegative Ricci curvature. As a consequence, we prove that it must be a finite quotient of $\mathbb {S}^4$, $\mathbb {CP}^2$, or $\mathbb {S}^3 \times \mathbb {R}$. In particular, a compact four-dimensional gradient shrinking soliton with pinched Weyl curvature $(*)$ must be $\mathbb {S}^4$, $RP^4$ or $\mathbb {CP}^2$.

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