arXiv 2103.112892021-03-23math.DG

ALF和ALG流形的正质量定理

Positive mass theorems of ALF and ALG manifolds

Peng Liu, Yuguang Shi, Jintian Zhu

备注：: 32pages, 5 figures, all comments are welcome!

摘要：

本文证明了模空间分别为$\mathbb R^{n-1}\times \mathbb S^1$和$\mathbb R^{n-2}\times \mathbb T^2$且维数不大于$7$的ALF和ALG流形的正质量定理（定理\ref{ALFPMT0}）. {不同于\cite [定理2]{minerbe 2008 a}中自旋结构的相容性条件，我们证明了对于$\mathbb S^1$和$\mathbb T^2$，某种类型的不可压缩条件足以保证质量的非负性。}在渐近平坦的情况下，我们减少了所需的正质量定理，这些关于不存在的正标量曲率度量的闭流形来自广义手术PDMATHAMT-环面。{最后，我们研究了某些填充问题，并得到了平坦积$2$ -环面$\mathbb S^1(l_1)\times \mathbb S^1(l_2)$的容许填充的总平均曲率的最优界。}

英文摘要：

In this paper, we want to prove positive mass theorems for ALF and ALG manifolds with model spaces $\mathbb R^{n-1}\times \mathbb S^1$ and $\mathbb R^{n-2}\times \mathbb T^2$ respectively in dimensions no greater than $7$ (Theorem \ref{ALFPMT0}). { Different from the compatibility condition for spin structure in \cite[Theorem 2]{minerbe2008a}, we show that some type of incompressible condition for $\mathbb S^1$ and $\mathbb T^2$ is enough to guarantee the nonnegativity of the mass.} As in the asymptotically flat case, we reduce the desired positive mass theorems to those ones concerning non-existence of positive scalar curvature metrics on closed manifolds coming from generalize surgery to $n$-torus. { Finally, we investigate certain fill-in problems and obtain an optimal bound for total mean curvature of admissible fill-ins for flat product $2$-torus $\mathbb S^1(l_1)\times \mathbb S^1(l_2)$.}

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