论文标题
标量曲率和平均曲率的变形
Deformation of the scalar curvature and the mean curvature
论文作者
论文摘要
在带有边界$ \ partial m $的紧凑型歧管$ m $上，我们研究了按$ m $开出标量曲率的问题，同时在边界$ \ partial m $上的平均曲率。为此，我们介绍了单数度量的概念，该概念是受Fischer-Marsden在[18]中的早期工作的启发，而[23]中的lin-Yuan in [23]的封闭歧管。我们表明，对于具有最小边界的通用标量流形歧管，我们可以同时规定标量曲率和平均曲率。我们还证明了具有完全测量边界的平面流形的一些刚度结果。
On a compact manifold $M$ with boundary $\partial M$, we study the problem of prescribing the scalar curvature in $M$ and the mean curvature on the boundary $\partial M$ simultaneously. To do this, we introduce the notion of singular metric, which is inspired by the early work of Fischer-Marsden in [18] and Lin-Yuan in [23] for closed manifold. We show that we can prescribe the scalar curvature and the mean curvature simultaneously for generic scalar-flat manifolds with minimal boundary. We also prove some rigidity results for the flat manifolds with totally geodesic boundary.
