Constraints are not enough

The Euclidean Einstein-Hilbert action is well-known to be unbounded below and thus to raise many questions regarding the definition of the gravitational path integral. A variety of works since the late 1980's have suggested that this problem disappears when one fixes a foliation of the spacetime and imposes the corresponding gravitational constraints. However, we show here that this approach fails with various classes of boundary conditions imposed on the foliation: compact slices without boundary, asymptotically flat, or asymptotically locally anti-de Sitter slices. We also discuss the idea of fixing the scalar curvature and Wick-rotating the conformal factor, and show that it also fails to produce an action bounded from below.