标题： 可缩流形上的正曲率条件 显示英文标题

标题： Positive curvature conditions on contractible manifolds

摘要： 我们的目标是确定曲率条件，以区分开的、可缩的流形中的欧几里得空间以及有边界的紧致、可缩流形中的圆盘。 首先，我们证明了一个足够连通的、紧致的、可缩的带边5-流形的内部，如果支持一个具有均匀正标量曲率的完备黎曼度量，则微分同胚于欧几里得5-空间。 接下来，我们研究了带边界的紧致流形的类似问题：一个支持具有正标量曲率的黎曼度量且边界为正平均曲率的紧致、可缩流形是否一定是圆盘？ 我们给出了示例，表明仅凭这种曲率条件无法区分圆盘；另一方面，我们展示了更强的曲率条件，使我们能够得出这样的结论。 显示英文摘要

摘要： Our goal is to identify curvature conditions that distinguish Euclidean space in the case of open, contractible manifolds and the disk in the case of compact, contractible manifolds with boundary. First, we show that an open manifold that is the interior of a sufficiently connected, compact, contractible 5-manifold with boundary and supports a complete Riemannian metric with uniformly positive scalar curvature is diffeomorphic to Euclidean 5-space. Next, we investigate the analogous question for compact manifolds with boundary: Must a compact, contractible manifold that supports a Riemannian metric with positive scalar curvature and mean convex boundary necessarily be the disk? We present examples demonstrating that this curvature condition alone cannot distinguish the disk; on the other hand, we exhibit stronger curvature conditions that allow us to draw such a conclusion.