发表机构
Southern Connecticut State University; University of Chicago（南康涅狄格州立大学; 芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在三维空间中回答了Gromov关于测度收敛下非负标量曲率保持的问题，证明无额外假设时极限度量可任意，但若恒等映射一致双Lipschitz则曲率下界保持，并推广至熵条件与L^p收敛情形。
AI 中文摘要
假设光滑度量 $g_k$ 在测度意义下收敛到光滑度量 $g$。Gromov 提出了如下问题：如果所有度量 $g_k$ 具有非负标量曲率，那么 $g$ 是否也必然具有非负标量曲率？我们在三维空间中回答了 Gromov 的问题。我们证明，在没有进一步假设的情况下，度量 $g$ 不必具有非负标量曲率，事实上 $g$ 可以是完全任意的。然而，如果额外假设恒等映射 $(M,g_k)\ o (M,g)$ 是一致双 Lipschitz 的，那么 $g$ 确实必须具有非负标量曲率。我们的证明方法还可用于证明，在几乎欧几里得熵条件下，当度量及其逆在 $L^p$（对适当大的 $p$）中收敛时，标量曲率下界将保持。
英文摘要
Assume that smooth metrics $g_k$ converge in measure to a smooth metric $g$. Gromov asked the following question: if all the metrics $g_k$ have non-negative scalar curvature does it follow that $g$ also has non-negative scalar curvature? We answer Gromov's question in dimension three. We show that, without further assumptions, the metric $g$ need not have non-negative scalar curvature and in fact $g$ may be completely arbitrary. However, if one assumes in addition that the identity maps $(M,g_k)\to (M,g)$ are uniformly bi-Lipschitz, then indeed $g$ must have non-negative scalar curvature. Our method of proof can also be used to show that, under an almost Euclidean entropy condition, scalar curvature lower bounds will persist under convergence of the metrics together with their inverses in $L^p$ for suitably large $p$.
Comments29 pages, comments are welcome!
