论文标题
宏观标量曲率和局部崩溃
Macroscopic scalar curvature and local collapsing
论文作者
论文摘要
考虑一个封闭的Riemannian $ n $ -Manifold $ m $，承认弯曲的Riemannian度量。我们表明，对于$ m $的每个Riemannian度量，$ m $的通用封面中有一个点，以使此时中心的每个半径$ r \ geq 1 $的体积更大或等于超质量$ n $ space中同一半径的体积。我们还根据宏观标量曲率对此结果进行解释。该结果在多面体长度空间的背景下更普遍地保持与Guth的问题有关。它的证明依赖于涉及Alexandrov/Urysohn宽度的度量几何进展的概括，该宽度涉及一定范围内半径的体积，并且在不同的尺度下崩溃。
Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume of every ball of radius $r \geq 1$ centered at this point is greater or equal to the volume of the ball of the same radius in the hyperbolic $n$-space. We also give an interpretation of this result in terms of macroscopic scalar curvature. This result, which holds more generally in the context of polyhedral length spaces, is related to a question of Guth. Its proof relies on a generalization of recent progress in metric geometry about the Alexandrov/Urysohn width involving the volume of balls of radius in a certain range with collapsing at different scales.
