The geometry of vacuum static spaces and deformations of scalar curvature

In this dissertation we mainly study the geometric structure of vacuum static spaces and some related geometric problems.;In particular, we have made progress in solving the classification problem of vacuum static spaces and in proving the Besse conjecture which is about manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3.;We also extend the local scalar curvature rigidity result of Brendle-Marques-Neves on upper hemisphere to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the work of Corvino. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work of Hang-Wang.;As for generic Riemannian manifolds, we find a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schrodinger type operator. In particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly conformal deformations which either increase or decrease scalar curvature. We find local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question arises naturally in the work of Corvino and Lohkamp.