Existence of static metrics with prescribed Bartnik boundary data

A static vacuum metric produces a Ricci flat manifold of one dimension higher and naturally arises on studying scalar curvature deformation and gluing. Originating from his quasi-local mass program in 1989, R. Bartnik conjectured that one can always find an asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. I will discuss well-posedness of this geometric boundary value problem and the recent progress toward the conjecture for large classes of boundary data. It is based on joint work with Zhongshan An.