The classical Bernstein theorem says that the only entire minimal graph in three-dimensional Euclidean space is a plane. Extensions to constant mean curvature (CMC) surfaces, higher dimensions, and curved ambient spaces have been a central theme in differential geometry for over a century.

This paper proves Bernstein-type theorems for CMC graphs in the three-dimensional light cone Q³₊ — a Lorentzian surface that arises naturally in conformal geometry and relativity. The result: if an entire CMC graph over the horosphere has Gaussian curvature bounded below, it must be either a horosphere or a sphere of Q³₊.

The light cone is degenerate — its induced metric has zero determinant — which makes differential geometry on it substantially harder than on Riemannian or even standard Lorentzian spaces. The Gaussian curvature bound from below is the key hypothesis: without it, exotic CMC graphs could exist. With it, the classification is complete — just two possibilities, horosphere or sphere. The rigidity result mirrors the Euclidean Bernstein theorem’s spirit (entire graphs are simple) but in a degenerate ambient geometry where “simple” means something different.