论文标题
BARTNIK数据的NNSC-Cobordism不存在
Nonexistence of NNSC-cobordism of Bartnik data
论文作者
论文摘要
在本文中，我们考虑了bartnik数据$（σ_1^{n-1}，γ_1，H_1）$和$（σ_2^{n-1}，γ_2，H_2）$的非负标量曲率（NNSC）的问题。我们证明，给定两个指标$γ_1$和$ s^{n-1} $上的$γ_2$（$ 3 \ le n \ le 7 $），$ h_1 $固定，然后$（s^{n-1}，γ_1，γ_1，h__1，h__1），h__1）$ and $（s^{n-1}，n no n n n n n n n n n n n sc，cob cob cob cob cob cob cob cob ersist cob ersist， $ h_2 $足够大（定理\ ref {highdimnoncob0}）。此外，我们表明，对于$ n = 3 $，一个弱的条件是，总平均曲率$ \ int_ {s^2}h_2dμ__{γ_2} $足够大，足够大的规则排除了NNSC COBORDISM（theorem \ ref Ref {2-D0}）;如果我们需要$γ_2$的高斯曲率为正，我们通过使用鹰质量和棕色York质量（定理\ ref {Cobordism20}）获得了不存在微不足道的NNSC-Cobordism的标准。 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}).
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}).
