arXiv 2403.025642024-03math.DG

具有夹挤尺度不变积分曲率的流形的长时间里奇流存在性与拓扑刚性

Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

The University of British Columbia（不列颠哥伦比亚大学）

Albert Chau, Adam Martens

备注：: Published version, in CAG. Minor changes only from previous version
发表信息：: Communications in Analysis and Geometry. Volume 34. Issue 1. 2026. Pages 285-333
DOI：: 10.4310/CAG.260509012855

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

证明了在曲率有界且尺度不变积分曲率相对于Sobolev常数足够夹挤的完备流形上，里奇流长时间存在；若曲率次临界L^p可积，则流形等距于标准平坦R^n，从而得到拓扑刚性。

摘要：

我们证明了从具有有界曲率和尺度不变积分曲率相对于其Sobolev常数的倒数足够夹挤的完备流形出发的里奇流长时间存在。此外，如果曲率是次临界$L^p$可积的，则该流局部光滑收敛到$M$上的极限度量$g(\infty)$，且$(M,g(\infty))$等距于标准平坦$\mathbb{R}^n$，这意味着$M$的拓扑刚性。这推广了Chen \cite{ChenEric}的工作，他证明了渐近平坦流形的类似结果。我们还证明了对于无界曲率初始数据（假设初始数据是如上所述的有界曲率流形的局部光滑极限）的长时间里奇流存在性（以及类似的拓扑刚性）结果。

英文摘要：

We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical $L^p$-integrable, this flow converges locally smoothly to a limiting metric $g(\infty)$ on $M$ with $(M,g(\infty))$ isometric to the standard flat $\mathbb{R}^n$, which implies topological rigidity of $M$. This generalizes work of Chen \cite{ChenEric}, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文