The study of static vacuum Riemannian metrics arises naturally in general relativity and differential geometry. A static vacuum metric produces a static spacetime by a warped product, and it is related to scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static vacuum metric with black hole boundary must belong to the Schwarzschild family. In contrast to the rigidity phenomenon, R. Bartnik conjectured that there are asymptotically flat, static vacuum metric realizing certain arbitrarily specified boundary data. I will discuss recent progress toward this conjecture. It is based on joint work with Zhongshan An.

In the second part of the talk (if time permits), I will discuss related topics in the non-time-symmetric case. In many ways, a stationary vacuum initial data set is a generalization of a static vacuum metric. However, from the point of view of deforming the dominant energy condition, we discover that a “ground state” initial data set need not be stationary vacuum but can sit in a null dust spacetime with a global Killing vector field, such as the pp-waves. This part is based on joint work with Dan Lee.