A De Lellis-Müller type estimate on the Minkowski lightcone (Q6583649)

It is well-known that any closed, embedded hypersurface in \(\mathbb{R}^{3}\) with vanishing trace-free part of the second fundamental form has constant mean curvature \(H\), and is thus a round sphere. There is also a quantitative version of this statement, due to \textit{C. de Lellis} and \textit{S. Müller} [J. Differ. Geom. 69, No. 1, 75--110 (2005; Zbl 1087.53004)], saying that if the trace-free part of the second fundamental form is sufficiently small in \(L^{2}\) , then the surface under consideration is \(W^{1,2}\)-close to a round sphere. \N\NIn a previous work, the author [Calc. Var. Partial Differ. Equ. 62, No. 3, Paper No. 90, 22 p. (2023; Zbl 1516.53082)] introduced a scalar second fundamental form \(A\) of space-like cross sections \(\Sigma\) of the standard Minkowski lightcone, such that the trace-free part of \(A\) vanishes if and only if \(\Sigma\) is a surface of constant spacetime mean curvature (STCMC) in the lightcone. Such a surface arises as an intersection with an affine, space-like hyperplane, and thus is precisely a round sphere up to a suitable Lorentz transformation. This paper gives a quantitative version of the aformentioned result in the spirit of De Lellis-Müller's work: the conformal factor \(\omega\) of a space-like cross section of the lightcone is close to the conformal factor of an appropiate STCMC surface in \(W^{2,2}\) if the trace-free part of \(A\) is sufficiently small in \(L^{2}\).