NNSC-Cobordism of Bartnik Data in High Dimensions
NNSC-Cobordism of Bartnik Data in High Dimensions
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DOI：
10.3842/sigma.2020.030
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发表时间：
2020-01
期刊：
影响因子：
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通讯作者：
Xue Hu;Yuguang Shi
中科院分区：
文献类型：
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作者：
Xue Hu;Yuguang Shi
在这篇短文中，我们提出了与非负标量曲率（NNSC）填充有关的三个问题。粗略地说，前两个问题集中在：$(n-1)$ -维Bartnik数据$\big(\Sigma_i ^{n-1}, \gamma_i, H_i\big)$, $i=1,2$， nnsc -协同？（即存在一个具有标量曲率$R(g)\geq 0$和边界$\partial \Omega=\Sigma_{1} \cup \Sigma_{2}$的$n$维紧致黎曼流形$\big(\Omega^n, g\big)$，其中$\gamma_i$是$g$在$\Sigma_i ^{n-1}$上的度规，$H_i$是$\big(\Omega^n, g\big)$中$\Sigma_i$的平均曲率）。如果$\big(\mathbb{S}^{n-1},\gamma_{\rm std},0\big)$与$\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$是正的标量曲率（PSC），其中$\big(\mathbb{S}^{n-1}, \gamma_{\rm std}\big)$表示标准圆单位球，则$\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$允许NNSC填充。正如格罗莫夫猜想与正质量定理有关一样，我们的问题与彭罗斯不等式有关，至少在$n=3$的情况下是这样。我们的第三个问题是关于$\Lambda\big(\Sigma^{n-1}, \gamma\big)$的，定义如下。
In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are $(n-1)$-dimensional Bartnik data $\big(\Sigma_i ^{n-1}, \gamma_i, H_i\big)$, $i=1,2$, NNSC-cobordant? (i.e., there is an $n$-dimensional compact Riemannian manifold $\big(\Omega^n, g\big)$ with scalar curvature $R(g)\geq 0$ and the boundary $\partial \Omega=\Sigma_{1} \cup \Sigma_{2}$ such that $\gamma_i$ is the metric on $\Sigma_i ^{n-1}$ induced by $g$, and $H_i$ is the mean curvature of $\Sigma_i$ in $\big(\Omega^n, g\big)$). If $\big(\mathbb{S}^{n-1},\gamma_{\rm std},0\big)$ is positive scalar curvature (PSC) cobordant to $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$, where $\big(\mathbb{S}^{n-1}, \gamma_{\rm std}\big)$ denotes the standard round unit sphere then $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$ admits an NNSC fill-in. Just as Gromov's conjecture is connected with positive mass theorem, our problems are connected with Penrose inequality, at least in the case of $n=3$. Our third problem is on $\Lambda\big(\Sigma^{n-1}, \gamma\big)$ defined below.