论文标题
与边界的四个manifolds上的共形刚性定理
Conformally invariant rigidity theorems on four-manifolds with boundary
论文作者
论文摘要
在文章中，我们介绍了具有边界的紧凑，定向的四个manifolds上的新的保形和平滑不变的。在第一部分中，我们表明，这些不变性的“阳性”条件将对具有边界的潜在歧管施加拓扑限制，这将M. Gursky M. M. Gursky和S.-Y. S.-Y的共同紧凑的Einstein Four-Manifolds的封闭的四个manifolds概括了结果。 A. Chang，J。Qing和P. Yang。在第二部分中，我们研究了Weyl在具有边界的四个manifolds上的功能，并建立了几个不变的刚性定理。作为应用，我们证明了与共同紧凑的爱因斯坦四拟合的一些刚性定理。这些结果将S.-Y.在封闭的四个manifolds上概括了这项工作。 A. Chang，J。Qing和P. Yang和P. Yang和刚性定理，用于G. Li，J。Qing和Y. Shi的同伴紧凑的爱因斯坦Four-Manifolds。证据的一个关键思想是了解边界附近平滑的riemannian度量的扩展。值得注意的是，我们排除了在具有脐带边界的歧管环境中研究封闭歧管引起的一些例子。
In the article we introduce new conformal and smooth invariants on compact, oriented four-manifolds with boundary. In the first part, we show that "positivity" conditions on these invariants will impose topological restrictions on underlying manifolds with boundary, which generalizes the results on closed four-manifolds by M. Gursky and on conformally compact Einstein four-manifolds by S.-Y. A. Chang, J. Qing, and P. Yang. In the second part, we study Weyl functional on four-manifolds with boundary and establish several conformally invariant rigidity theorems. As applications, we prove some rigidity theorems for conformally compact Einstein four-manifolds. These results generalize the work on closed four-manifolds by S.-Y. A. Chang, J. Qing, and P. Yang and rigidity theorem for conformally compact Einstein four-manifolds by G. Li, J. Qing, and Y. Shi. A crucial idea of the proofs is to understand the expansion of a smooth Riemannian metric near the boundary. It is noteworthy to point out that we rule out some examples arising from the study of closed manifolds in the setting of manifolds with umbilic boundary.
