Conference Agenda

In this note we investigate a notion of stability for spacelike cross sections of a null cone under area preserving variations that has been introduced in previous work by Kr\"oncke and the author. Here, we consider null cones with spherical cross sections in a $4$-dimensional spacetime and show that the Hawking energy of a stable cross section admits a non-negative lower bound provided the dominant energy condition holds. Similar to a recent work by Penuela Diaz, we show that under an additional assumption the Hakwing energy is zero if and only if the stable cross sections embeds isometrically into the Minkowski lightcone. As a main result, we show that the only stable cross sections of the standard Minkowski lightcone are round spheres.

Within general relativity, one of the most fundamental properties that any quasi-local energy is expected to satisfy in order to be deemed physically reasonable is the correct asymptotic behavior in the small sphere limit, first established by Horowitz and Schmidt for the Hawking energy in 1982. Given a spacetime $(M,\bar g)$, a point $p\in M$, and a future-directed timelike unit vector $e_0\in T_pM$, this limit is obtained by foliating the local future lightcone at $p$ by spacelike cross sections and evaluating the quasi-local energy along these sections as they approach the vertex.
Remarkably, although the small sphere limit has since been studied for many different notions of quasi-local energy, the underlying choice of background foliation has essentially always been the same: a foliation by constant-radius cuts in normal coordinates centered at the vertex. We refer to this distinguished choice as the natural background foliation. The purpose of this talk is to investigate the role of this foliation in more detail and study precisely how the small sphere limit of the Hawking energy depends on the choice of background foliation.
Surprisingly, this dependence turns out to be quite strong. After establishing the necessary null-geometric framework, we derive a curvature-dependent integral condition characterizing when, for a fixed tuple $(M,\bar g,p,e_0)$, a given background foliation yields the correct small sphere limit. However, if the small sphere limit is required to hold independently of the tuple $(M,\bar g,p,e_0)$, then the foliation is strongly constrained: it must, in a precise asymptotic sense, be $H^1$-close to the natural background foliation. In particular, among foliations whose leaves are obtained by linear rescalings along the null generators, the natural background foliation is the only one that yields the correct small sphere limit for all choices of $(M,\bar g,p,e_0)$.

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.

Surfaces of constant spacetime (or co-dimension $2$) mean curvature (STCMC) have been shown to be abundant in the asymptotic end(s) of any asymptotically Minkowskian spacetime of non-vanishing mass (C.—Sakovich ’21). It has since been an open question how many STCMC-surfaces there are in Minkowski spacetime. We will explain why there are in fact many STCMC-surfaces in Minkowski spacetime. Our analysis is based on a characterization of local STCMC-foliations in relativistic initial data sets by Metzger—Pe\~nuela which in turn goes back to a local CMC-foliation result in Riemannian manifolds by Ye.

We will also briefly touch on the corresponding Riemannian result by Yau who argues for rigidity of co-dimension $2$ constant mean curvature (CMC) surfaces in Euclidean $4$-space. In particular, we will indicate how the stark difference between the Euclidean and Minkowskian results can be resolved.