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 $\lambda_1 + \lambda_2\ge c_0 R>0$ everywhere for some $c_0\approx 0.29167$, where $\{\lambda_i\}$ are the two least eigenvalues of Ricci curvature. Furthermore, we will show that $\lambda_1 + \lambda_2\ge \frac 13R>0$ under a better pinched Weyl tensor assumption. We point out that the lower bound $\frac 13R$ is sharp.

中文翻译：

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四维梯度收缩孤子的间隙定理

在本文中，我们将证明四维梯度收缩孤子的间隙定理。更准确地说，我们将证明任何具有非负和有界 Ricci 曲率的完整四维梯度收缩孤子，满足收缩的 Weyl 曲率，要么是平坦的，要么是 $\lambda_1 + \lambda_2\ge c_0 R>0$ 对于某些 $ c_0\approx 0.29167$，其中 $\{\lambda_i\}$ 是 Ricci 曲率的两个最小特征值。此外，我们将证明 $\lambda_1 + \lambda_2\ge \frac 13R>0$ 在更好的收缩 Weyl 张量假设下。我们指出下限 $\frac 13R$ 是尖锐的。