论文标题
在带有体积和整体曲率边界的Riemannian歧管的极限上
On limit spaces of Riemannian manifolds with volume and integral curvature bounds
论文作者
论文摘要
在没有Apriori非碰撞假设下研究了具有L^P曲率边界的Riemannian流形的限制空间的规律性。由限制度量的局部体积生长条件定义的常规子集被证明具有riemannian歧管的结构。结果之一是具有$ l^p $曲率边界的Riemannian流形的紧凑定理，以及尖头的Cheeger-Gromov拓扑中的先验体积增长假设。还研究了不同的收敛概念，这用尖头的Cheeger-Gromov拓扑结构的球替代了疲惫，并用体积非collap的区域耗尽了。假设在RICCI曲率上有下限，则将紧凑性定理扩展到该拓扑。此外，我们研究了歧管的收敛序列如何在极限上拓扑上断开连接。在两个维度的基础上，详细描述了极限空间的结构：它被认为是不完整的Riemannian表面和一维长度空间的结合。
The regularity of limit spaces of Riemannian manifolds with L^p curvature bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is shown to carry the structure of a Riemannian manifold. One consequence of this is a compactness theorem for Riemannian manifolds with $L^p$ curvature bounds and an a priori volume growth assumption in the pointed Cheeger-Gromov topology. A different notion of convergence is also studied, which replaces the exhaustion by balls in the pointed Cheeger-Gromov topology with an exhaustion by volume non-collapsed regions. Assuming in addition a lower bound on the Ricci curvature, the compactness theorem is extended to this topology. Moreover, we study how a convergent sequence of manifolds disconnects topologically in the limit. In two dimensions, building on results of Shioya, the structure of limit spaces is described in detail: it is seen to be a union of an incomplete Riemannian surface and 1-dimensional length spaces.
