四维至七维的二次标量曲率衰减与一致正性
Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven
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中文总结 AI 辅助
针对四维至七维满足渐近二次标量曲率系数大于(n-1)/n的正标量曲率完备非紧黎曼流形，证明其存在标量曲率至少为1的完备光滑度量，证实了Gromov临界衰减速率猜想的对应部分。
中文摘要 AI 辅助
设4≤n≤7，(Mⁿ,g)为正标量曲率的完备、连通、可定向非紧黎曼流形。我们证明：若g的渐近二次标量曲率系数大于(n-1)/n，则M存在标量曲率至少为1的完备光滑度量。阈值(n-1)/n与严格不等式是最优的，这证实了Gromov提出的临界衰减速率猜想在四维至七维的第二部分。
英文摘要
Let $4\le n\le7$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold of positive scalar curvature. We prove that if the asymptotic quadratic scalar curvature coefficient of $g$ is greater than $(n-1)/n$, then $M$ carries a complete smooth metric whose scalar curvature is at least one. The threshold $(n-1)/n$ and the strict inequality are optimal. This confirms the second part of Gromov's critical rate of decay conjecture in dimensions four through seven.
发表机构
- School of Sciences, Great Bay University（大湾区大学理学院）
- Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究所）
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