We prove that a 4-dimensional \(C^2\) conformally compact Einstein manifold with Hölder continuous scalar curvature and with \(C^{m,\alpha }\) boundary metric has a \(C^{m,\alpha }\) compactification. We also study the regularity of the new structure and the new defining function. This is a supplementary proof of Anderson’s work and an improvement of Helliwell’s result in dimension 4.

中文翻译：

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4维共形紧的爱因斯坦流形的有限边界正则性

我们证明具有Hölder连续标量曲率和\（C ^ {m，\ alpha} \）边界度量的4维\（C ^ 2 \）保形紧致的爱因斯坦流形具有\（C ^ {m，\ alpha } \）压缩。我们还研究了新结构和新定义功能的规律性。这是对安德森工作的补充证明，也是对维利结果在维度4中的改进。