Let (M, g) be a compact Riemannian n-dimensional manifold with umbilic boundary It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have \(\partial M\) as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is \(n=6,7,8\), provided that the Weyl tensor is always not vanishing on the boundary.

中文翻译：

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具有脐带边界的低维流形上标量平坦度量的紧致性结果

令 ( M ,  g ) 是具有脐带边界的紧凑黎曼n维流形 众所周知，在某些假设下，在g的共形类中存在具有\(\partial M\)作为恒定平均曲率超曲面。在本文中，我们证明这些度量在低维流形的情况下是一个紧集，即\(n=6,7,8\)，前提是 Weyl 张量始终不会在边界上消失。