Markus Wolff: A De Lellis--Mueller type estimate for surfaces on the lightcone

We define a timelike \(4\)-vector associated to a spacelike cross section of the standard Minkowski lightcone in \(3+1\)-dimensions that transforms covariantly under Lorentz boosts of the ambient spacetime, and discuss its connection with a notion of center of mass in the conformal class of the standard \(2\)-sphere. In particular, any surface of constant curvature embedded into the lightcone corresponds uniquely to a timelike \(4\)-vector that fully determines the surface.

As an application, we obtain a quantitive \(W^{2,2}\)-estimate between a cross section of the standard lightcone and a surface of constant curvature corresponding to the associated timelike \(4\)-vector of the cross section depending on the \(L^2\)-norm of the tracefree part of a scalar valued notion of second fundamental form of the cross section along the lightcone, similar to work by De Lellis--Mueller in Euclidean space.