Area-constrained Willmore surfaces in Lorentzian manifolds: curvature inequalities and the Hawking quasi-local energy

RESUMO

We develop an area-constrained Willmore theory for closed spacelike surfaces in four-dimensional Lorentzian manifolds. Since the Euler--Lagrange equation is normal-bundle-valued, we introduce directional criticality along a prescribed normal direction and identify a class of connection-compatible directions determined by the normal connection. Our main result is the sharp inequality $2\lambda|\Sigma|+\int_\Sigma |\vec H|^2,d\mu\leq 8\pi\chi(\Sigma)$ for criticality in a connection-compatible spacelike direction under an Einstein-tensor sign condition. Under the dominant energy condition and $\lambda\geq0$, this yields $\int_\Sigma |\vec H|^2,d\mu\leq16\pi$. Equality is rigid: $\Sigma$ is round and satisfies Minkowski causal-diamond rigidity. We also prove an analogous sharp inequality for null critical directions. These results give positivity, rigidity, stationarity, and infinitesimal monotonicity statements for the Hawking quasi-local energy. A hypersurface formulation recovers the previously studied Hawking-surface problem as criticality in the spatial normal direction. Explicit examples in spherical symmetry, Kerr, and FLRW spacetimes distinguish directional from full criticality and show the necessity of the sign assumptions in the rigidity theory.