S.-Y. A. Chang, J. Qing, and P. Yang proved an important gap theorem for Bach-flat metrics with round sphere as model case in 2007. In this article, we generalize this result by establishing conformally invariant gap theorems for Bach-flat 4-manifolds with \((\mathbb {CP}^2, g_{FS})\) and \((S^2 \times S^2,g_{prod})\) as model cases. An iteration argument plays an important role in the case of \((\mathbb {CP}^2, g_{FS})\) and the convergence theory of Bach-flat metrics is of particular importance in the case of \((S^2 \times S^2,g_{prod})\). The latter result provides an interesting way to distinguish \((S^2 \times S^2,g_{prod})\) from \((\mathbb {CP}^2\#\bar{\mathbb {CP}}^2,g_{Page})\).

S.-YA Chang、J.Q.Qing 和 P. Yang 在 2007 年证明了一个重要的 Bach-flat 度量的间隙定理，以圆球为模型案例。在本文中，我们通过为 Bach 建立保形不变间隙定理来推广这一结果 -以\((\mathbb {CP}^2, g_{FS})\)和\((S^2 \times S^2,g_{prod})\)作为模型案例的平面 4-流形。迭代参数在\((\mathbb {CP}^2, g_{FS})\)的情况下起着重要作用，而 Bach-flat 度量的收敛理论在\((S ^2 \times S^2,g_{prod})\)。后一个结果提供了一种有趣的方式来区分\((S^2 \times S^2,g_{prod})\)和\((\mathbb {CP}^2\#\bar{\mathbb {CP}} ^2,g_{Page})\).