Abstract
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 \((\mathbb {CP}^2\#\bar{\mathbb {CP}}^2,g_{Page})\).
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Acknowledgements
The author would like to thank his advisor Professor Sun-Yung A. Chang for her constant support. Thanks also to Professor Matthew Gursky for numerous helpful comments and enlightening discussions.
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Communicated by A. Malchiodi.
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