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 in three-dimensional manifolds with scalar curvature bounds. Building on the Christodoulou-Yau inequality $H^2\le 16\pi \mathrm{/}\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$ (with $H$ the mean curvature and $\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$ the area), we show that the associated rigidity phenomena persist under a weaker notion of stability controlling only the constant mode of the second variation, combined with an extrinsic curvature sign condition. This yields Euclidean rigidity without imposing intrinsic symmetry or near-roundness assumptions and extends to higher dimensions and to the hyperbolic and spherical settings. In the Lorentzian setting we consider spacetime constant mean curvature (STCMC) surfaces, a natural generalization of CMC surfaces. We introduce a stability theory for STCMC surfaces and prove the sharp inequality $\mathrm{\mid }\vec{H}{\mathrm{\mid }}^2\le 16\pi \mathrm{/}\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$ under the dominant energy condition. We also obtain rigidity for the equality case: under suitable geometric assumptions the surface is intrinsically round and the spacetime region it bounds is flat, with maximal globally hyperbolic development isometric to a causal diamond in Minkowski spacetime. Finally, we show that the canonical asymptotic STCMC foliations known in both the spacelike and null settings have leaves that are stable with respect to this notion of stability.