论文标题
Bartnik数据的NNSC-bobordism在高维
NNSC-Cobordism of Bartnik Data in High Dimensions
论文作者
论文摘要
在简短的说明中，我们制定了与非负标量曲率（NNSC）填充有关的三个问题。宽松地说，前两个问题的重点是：$（n-1）$ - 尺寸bartnik data $ \ big（σ_i ^{n-1}，γ_i，γ_i，h_i \ big）$，$ i = 1,2 $，nnsc-cobordant？ （即，有一个$ n $维的紧凑型riemannian歧管$ \ big（ω ^n，g \ big）$，带有标量曲率$ r（g）\ geq 0 $和边界$ \ partial $ \ partialω=σ_{1} {1} \cupσ_{2} $ n是$ n $ usguse $ n是$ g n是$ g n是$ g n是$γ_i ^ $ g $，$ h_i $是$ \ big（ω^n，g \ big）$的$σ_i$的平均曲率）。如果$ \ big（\ mathbb {s}^{n-1}，γ_ {\ rm std}，0 \ big）$是正标度曲率（PSC）cobordant to $ \ big（σ_1^{n-1^{n-1}，γ_1，γ_1，γ_1，H_1 \ big）$，$ \ big \ big（s） γ_ {\ rm std} \ big）$表示标准的圆形单位球，然后$ \ big（σ_1 ^{n-1}，γ_1，H_1 \ big）$ $ nnsc填充。正如格罗莫夫（Gromov）的猜想与正质量定理有关，至少在$ n = 3 $的情况下，我们的问题也与Penrose不平等相关。我们的第三个问题是在下面定义的$λ\ big（σ^{n-1}，γ\ big）$上。
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 $\big(Σ_i ^{n-1}, γ_i, H_i\big)$, $i=1,2$, NNSC-cobordant? (i.e., there is an $n$-dimensional compact Riemannian manifold $\big(Ω^n, g\big)$ with scalar curvature $R(g)\geq 0$ and the boundary $\partial Ω=Σ_{1} \cup Σ_{2}$ such that $γ_i$ is the metric on $Σ_i ^{n-1}$ induced by $g$, and $H_i$ is the mean curvature of $Σ_i$ in $\big(Ω^n, g\big)$). If $\big(\mathbb{S}^{n-1},γ_{\rm std},0\big)$ is positive scalar curvature (PSC) cobordant to $\big(Σ_1 ^{n-1}, γ_1, H_1\big)$, where $\big(\mathbb{S}^{n-1}, γ_{\rm std}\big)$ denotes the standard round unit sphere then $\big(Σ_1 ^{n-1}, γ_1, H_1\big)$ 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 $Λ\big(Σ^{n-1}, γ\big)$ defined below.
