Conformally invariant rigidity theorems on four-manifolds with boundary

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.