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A gap theorem of four-dimensional gradient shrinking solitons

In this paper, we will prove a gap theorem for 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 $λ_1 + λ_2\ge c_0 R>0$ everywhere for some $c_0\approx 0.29167$, where $\{λ_i\}$ are the two least eigenvalues of Ricci curvature. Furthermore, we will show that $λ_1 + λ_2\ge \frac 13R>0$ under a better pinched Weyl tensor assumption. We point out that the lower bound $\frac 13R$ is sharp.