Let $ (M,g) $ a compact Riemannian $ n $-dimensional manifold. 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. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term $ h_{g} $ with a negative smooth function $ \alpha, $ the set of solutions of Yamabe problem is still a compact set.

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具有边界的流形上线性摄动山部问题的紧致性结果

设（M，g）$紧凑的黎曼$ n $维流形。众所周知，在某些假设下，在$ g $的保形类中，存在标量平坦的度量，其中$ \ partial M $作为恒定平均曲率超曲面。同样，在某些假设下，已知这些度量是紧凑集合。本文证明，在脐带边界和非脐带边界的情况下，如果我们用负光滑函数$ \ alpha线性扰动平均曲率项$ h_ {g} $，则Yamabe问题的解集仍然是一个紧凑的组合。