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Spacelike hypersurfaces in the Lorentz-Minkowski (n+1)dimensional space L can be endowed with another Riemannian metric, the one induced by the Euclidean space R. The hypersurfaces with the same mean curvature with respect to both metrics can be locally determined by a smooth function u satisfying |Du| < 1, and being the solution to a certain partial differential equation. We call this equation the HR = HL hypersurface equation. In the particular case in which n = 2 and both curvatures vanish, Kobayashi proved that the graphs determined by the solutions of such equation are open pieces of spacelike planes or helicoids, in the region where they are spacelike. In this manuscript we prove the existence of a family of solutions whose graphs have non-zero mean curvature, and we present an inequality relating the mean curvature to the width of the domain of certain solutions, those without critical points.

On the Solutions to the $H_R H_L$ Hypersurface Equation