Necessity and Extensions of Gibbons--Hawking--York Boundary Terms: Variational Well-Posedness, Corners and Null Boundaries, and Closure to Quasilocal Energy and Thermodynamics

On pseudo-Riemannian manifolds with (possibly non-smooth) boundaries, the variation of the Einstein--Hilbert bulk action contains normal derivative-type boundary fluxes; fixing only Dirichlet data for the induced metric h_{ab} does not suffice for well-posedness. In the framework of Levi--Civita connection and extrinsic curvature, this paper rigorously proves that adding the Gibbons--Hawking--York (GHY) term with orientation factor \varepsilon:=n^\mu n_\mu\in\{\pm 1\} at non-null boundaries canc