Curvature inequalities and rigidity for constant mean curvature and spacetime constant mean curvature surfaces

We establish curvature inequalities and rigidity results for surfaces satisfying constant mean curvature type conditions in both Riemannian and Lorentzian geometry. In the Riemannian setting, we study constant mean curvature (CMC) surfaces. Building on the Christodoulou-Yau inequality $H^2\leq 16\pi / |\Sigma|$ (with $H$ the mean curvature and $|\Sigma |$ the area) for CMC surfaces on three-dimensional manifolds with nonnegative scalar curvature, we show that the inequality holds under a weaker stability condition controlling only the constant mode of the second variation. Combined with an extrinsic curvature sign condition, equality forces the region enclosed by the surface to be Euclidean. These results extend to higher dimensions and to the hyperbolic and spherical settings.
In the Lorentzian setting, we introduce a stability theory for spacetime constant mean curvature (STCMC) surfaces and prove the sharp inequality $|\vec{H}|^2\leq 16\pi / |\Sigma|$ under the dominant energy condition. We also obtain rigidity for the equality case: under suitable geometric assumptions, the maximal globally hyperbolic development of the enclosed spacelike region is isometric to a causal diamond in Minkowski spacetime. In particular, this implies positivity and rigidity for the Hawking quasi-local energy in the general spacetime setting when evaluated on stable STCMC surfaces. Finally, we analyze the known STCMC foliations in the spacelike and null settings. We show that asymptotic leaves are stable under positive mass conditions, whereas the local matter density and shear govern the instability of local foliations.