A De Lellis-M\"uller type estimate on the Minkowski lightcone

We prove an analogue statement to an estimate by De Lellis-M\"uller in $\mathbb{R}^3$ on the standard Minkowski lightcone. More precisely, we show that under some additional assumptions, any spacelike cross section of the standard lightcone is $W^{2,2}$-close to a round surface provided the trace-free part of a scalar second fundamental form $A$ is sufficiently small in $L^2$. To determine the correct intrinsically round cross section of reference, we define an associated $4$-vector, which transforms equivariantly under Lorentz transformations in the restricted Lorentz group. A key step in the proof consists of a geometric, scaling invariant estimate, and we give two different proofs. One utilizes a recent characterization of singularity models of null mean curvature flow along the standard lightcone by the author, while the other is heavily inspired by an almost-Schur lemma by De Lellis-Topping.