具有Ricci曲率下界的度量的环面与正质量稳定性
Torus and Positive Mass stability for metrics with Ricci curvature lower bound
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中文总结 AI 辅助
该研究证明了满足第一稳定systole下界、Ricci曲率下界、直径上界且标量曲率负部$L^1$范数消失的环面度量序列，其子序列会收敛到环面平坦度量；类似地，满足Ricci曲率下界、非负标量曲率且ADM质量消失的渐近平坦自旋黎曼流形序列，会收敛到欧氏空间。
中文摘要 AI 辅助
考虑环面上的一列度量$g_i$，其成员满足第一稳定 systole 的一致下界、Ricci曲率的一致下界，以及直径的一致上界。若沿该序列，标量曲率负部的$L^1$范数消失，则我们证明存在该度量的一个子序列，在测地-Gromov-Hausdorff拓扑下收敛到环面上的平坦度量。对于渐近平坦的自旋黎曼流形序列，若其满足Ricci曲率下界、非负标量曲率，且沿序列的ADM质量消失，则存在类似结论：该序列会在点测地Gromov-Hausdorff拓扑下收敛到欧氏空间。
英文摘要
Consider a sequence of metrics $g_i$ on the torus whose members have uniform lower bounds on their first stable systoles and Ricci curvatures, and have a uniform upper bound on their diameters. If the $L^1$ norm of the negative part of the scalar curvatures vanishes along this sequence, then we show there is a subsequence of the metrics which converges in the measured-Gromov-Hausdorff topology to a flat metric on the torus. Something analogous holds for sequences of asymptotically flat spin Riemannian manifolds with a uniform lower bound on Ricci curvature and nonnegative scalar curvature: if the ADM masses of the distinguished ends tend to zero along the sequence, then the manifolds converge to Euclidean space in the pointed measured Gromov-Hausdorff sense.
发表机构
- University of Antwerp（安特卫普大学）
- Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。
