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Compactness results for linearly perturbed Yamabe problem on manifolds with boundary
Discrete and Continuous Dynamical Systems-Series S ( IF 1.3 ) Pub Date : 2020-11-16 , DOI: 10.3934/dcdss.2020453
Marco Ghimenti , , Anna Maria Micheletti

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.

中文翻译:

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。
更新日期:2020-11-16
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