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Some conformally invariant gap theorems for Bach-flat 4-manifolds
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-07-02 , DOI: 10.1007/s00526-021-02022-5
Siyi Zhang 1
Affiliation  

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 \((\mathbb {CP}^2\#\bar{\mathbb {CP}}^2,g_{Page})\).



中文翻译:

Bach-flat 4-manifolds的一些共形不变间隙定理

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\((\mathbb {CP}^2, g_{FS})\)\((S^2 \times S^2,g_{prod})\)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\((S^2 \times S^2,g_{prod})\)\((\mathbb {CP}^2\#\bar{\mathbb {CP}} ^2,g_{Page})\).

更新日期:2021-07-02
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