论文标题
标量曲率和测量球的相对能力
Scalar curvature and the relative capacity of geodesic balls
论文作者
论文摘要
在Riemannian歧管中，众所周知，可以从小的大地球（球形）的体积（区域）中回收标量曲率。我们显示标量曲率同样由同心小测量球的相对能力确定。该结果具有从一般相对论的动机（作为对大球在渐近平坦的多种歧管中的能力的先前研究的补充）和非负标量曲率的弱定义。它还激发了一个猜想（灵感来自灰色和范海克的著名猜想），关于小规模上相对容量的类似欧几里得的行为是否足以将空间表征为平坦。
In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes (areas) of small geodesic balls (spheres). We show the scalar curvature is likewise determined by the relative capacities of concentric small geodesic balls. This result has motivation from general relativity (as a complement to a previous study by the author of the capacity of large balls in an asymptotically flat manifold) and from weak definitions of nonnegative scalar curvature. It also motivates a conjecture (inspired by the famous volume conjecture of Gray and Vanhecke), regarding whether Euclidean-like behavior of the relative capacity on the small scale is sufficient to characterize a space as flat.
