A De Lellis–Müller type estimate on the Minkowski lightcone

Abstract

We prove an analogue statement to an estimate by De Lellis–Müller 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.