A gap theorem of four-dimensional gradient shrinking solitons

In this paper, we will prove a gap theorem on four-dimensional gradient shrinking soliton. More precisely, we will show that any complete four-dimensional gradient shrinking soliton with nonnegative and bounded Ricci curvature, satisfying a pinched Weyl curvature, either is flat, or $\lambda_1 + \lambda_2 \geq c_0 R \gt 0$ at all points, where $c_0 \approx 0.29167$ and $\lbrace \lambda_i \rbrace$ are the two least eigenvalues of Ricci curvature. Furthermore, we can improve our estimate to $\lambda_1 + \lambda_2 \geq \frac{1}{3} R \gt 0$ under a stronger pinched condition. We point out that the lower bound $\frac{1}{3} R$ is sharp.

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Received 4 June 2016

Accepted 4 February 2018

Published 6 July 2020