I will discuss sharp curvature inequalities and rigidity results for surfaces satisfying constant mean curvature type conditions in both Riemannian and Lorentzian geometry.
In the Riemannian setting, I will consider closed CMC surfaces in three-dimensional manifolds with scalar curvature lower bounds. I will explain how the Christodoulou–Yau inequality $H^2 \leq 16\pi / |\Sigma|$ extends to a weaker notion of stability controlling only the constant mode of the second variation. Combined with a sign condition on the ambient Ricci curvature, this yields rigidity in the equality case without assuming intrinsic symmetry or near-roundness. I will also briefly discuss analogous results in the hyperbolic and spherical settings.
In the Lorentzian setting, I will introduce a notion of stability for spacetime constant mean curvature (STCMC) surfaces and present the sharp inequality $|\vec{H}|^2 \leq 16\pi / |\Sigma|$ under the dominant energy condition. I will then describe the corresponding rigidity statement in the equality case and its relation to Hawking quasi-local mass. This talk is based on the preprint arXiv:2603.16707.