Bartnik数据的NNSC协边性的不存在性
Nonexistence of NNSC-cobordism of Bartnik data
摘要：
本文考虑Bartnik数据$(Σ_1^{n-1}, γ_1, H_1)$和$(Σ_2^{n-1}, γ_2, H_2)$的非负数量曲率（NNSC）配边问题.我们证明了给定两个度量$γ_1$和$γ_2$在$S^{n-1}$（$3\le n\le 7$）上，且$H_1$是固定的，那么只要规定的平均曲率$H_2$足够大，则$(S^{n-1}, γ_1, H_1)$和$(S^{n-1}, γ_2, H_2)$不允许NNSC协边性（定理\ref{highdimnoncob0}）.此外，我们证明了对于$n=3$，一个更弱的条件，即总平均曲率$\int_{S^2}H_2dμ_{γ_2}$足够大，排除了NNSC配边（定理\ref{2-d0}）;如果我们要求$γ_2$的高斯曲率为正，我们得到了一个标准的平凡NNSC配边不存在通过使用霍金质量和布朗约克质量（定理\ref{cobordism20}）。对于一般的拓扑情形，证明了在给定的平均曲率$H_2$足够大的条件下，$(Σ_1^{n-1}, γ_1, 0)$和$(Σ_2^{n-1}, γ_2, H_2)$不允许NNSC配边（定理\ref{highdimnoncob10}）.
英文摘要：
In this paper, we consider the problem of nonnegative scalar curvature (NNSC) cobordism of Bartnik data $(Σ_1^{n-1}, γ_1, H_1)$ and $(Σ_2^{n-1}, γ_2, H_2)$. We prove that given two metrics $γ_1$ and $γ_2$ on $S^{n-1}$ ($3\le n\le 7$) with $H_1$ fixed, then $(S^{n-1}, γ_1, H_1)$ and $(S^{n-1}, γ_2, H_2)$ admit no NNSC cobordism provided the prescribed mean curvature $H_2$ is large enough(Theorem \ref{highdimnoncob0}). Moreover, we show that for $n=3$, a much weaker condition that the total mean curvature $\int_{S^2}H_2dμ_{γ_2}$ is large enough rules out NNSC cobordisms(Theorem \ref{2-d0}); if we require the Gaussian curvature of $γ_2$ to be positive, we get a criterion for non existence of trivial NNSC-cobordism by using Hawking mass and Brown-York mass(Theorem \ref{cobordism20}). For the general topology case, we prove that $(Σ_1^{n-1}, γ_1, 0)$ and $(Σ_2^{n-1}, γ_2, H_2)$ admit no NNSC cobordism provided the prescribed mean curvature $H_2$ is large enough(Theorem \ref{highdimnoncob10}).
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