We consider the problem of finding on a given Euclidean domain \(\Omega \) of dimension \(n \ge 3\) a complete conformally flat metric whose Schouten curvature A satisfies some equations of the form \(f(\lambda (-A)) = 1\). This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary \(\partial \Omega \) is a smooth bounded hypersurface (of codimension one). When \(\partial \Omega \) contains a compact smooth submanifold \(\Sigma \) of higher codimension with \(\partial \Omega {\setminus }\Sigma \) being compact, we also give a ‘sharp’ condition for the divergence to infinity of the conformal factor near \(\Sigma \) in terms of the codimension.

中文翻译：

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Loewner-Nirenberg问题的完全非线性版本的存在性和唯一性

我们考虑在给定的尺寸\（n \ ge 3 \）的欧几里得域\（\ Omega \）上找到一个完整的保形平坦度量的问题，其Schouten曲率A满足一些形式为\（f（\ lambda（- A））= 1 \）。这概括了Loewner和Nirenberg考虑的标量曲率问题。当边界\（\ partial \ Omega \）是光滑的有界超曲面（维数为一）时，我们证明了这种度量的存在性和唯一性。当\（\ partial \ Omega \）包含具有更高维数的紧致光滑子流形\（\ Sigma \）和\（\ partial \ Omega {\ setminus} \ Sigma \）如果是紧实的，我们还给出了一个“尖锐”的条件，即就余维而言，在\（\ Sigma \）附近的保形因子发散到无穷大。