非负曲率三维流形的体积增长与积分曲率界
Volume growth and integral curvature bound for non-negatively curved three-manifolds
- National Tsing Hua University（国立清华大学）
- The Chinese University of Hong Kong（香港中文大学）
- The Hong Kong Polytechnic University（香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究三维非负曲率流形，证明欧氏体积增长等价于平均二次曲率衰减，并给出基于标量曲率积分阈值的判定及间隙定理。
AI中文摘要：
受凯勒几何中结果的启发，本文关注在三维非负曲率条件下，积分曲率界与体积增长之间的关系。在非负截面曲率情形下，我们证明：对于欧氏空间上的度量，具有欧氏体积增长当且仅当具有平均二次曲率衰减。这一结论基于证明：三维欧氏空间上具有非负截面曲率的度量，若其标量曲率的渐近缩放不变积分小于尖锐常数$8\pi$，则具有欧氏体积增长。此外，在非负里奇曲率条件下，若曲率在平均意义下衰减足够快，我们还证明了一个间隙定理。
英文摘要：
Motivated by results in Kähler geometry, in this work, we are interested in understanding the relation between integral curvature bounds and volume growth, under non-negative curvature in dimension three. In case of non-negative sectional curvature, we show that for metric on Euclidean space, it is of Euclidean volume growth if and only if it has average quadratic curvature decay. This is based on showing that metrics on three-dimensional Euclidean space with non-negative sectional curvature is of Euclidean volume growth if its asymptotic scaling invariant integral of scalar curvature is smaller than the sharp constant $8π$. We also show a gap Theorem if the curvature decay fast enough in the average sense, under non-negative Ricci curvature.
