A note on the stability of surfaces along null cones under area-preserving variations

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 Kröncke and the author. Here, we consider null cones, i.e., null hypersurfaces with non-vanishing expansion, with spherical cross sections in a 4-dimensional spacetime and show that the Hawking energy of a stable cross section admits a nonnegative lower bound provided the dominant energy condition holds. Similar to a recent work by Peñuela Diaz, we show that under an additional assumption the Hawking energy is zero if and only if the stable cross section 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.