Some conformally invariant gap theorems for Bach-flat 4-manifolds

Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat $4$-manifolds with $({\mathbb{C}\mathbb{P}}^2,g_{FS})$ and $({\mathbb{S}}^2\times {\mathbb{S}}^2,g_{prod})$ as model cases. An iteration argument plays an important role in the case of $({\mathbb{C}\mathbb{P}}^2,g_{FS})$ and the convergence theory of Bach-flat metrics is of particular importance in the case of $({\mathbb{S}}^2\times {\mathbb{S}}^2,g_{prod})$.