Geometric properties of surfaces with the same mean curvature in R3R3 and L3L3

Spacelike surfaces in the Lorentz–Minkowski space L3L3 can be endowed with two different Riemannian metrics, the metric inherited from L3L3 and the one induced by the Euclidean metric of R3R3. It is well known that the only surfaces with zero mean curvature with respect to both metrics are open pieces of the helicoid and of spacelike planes. We consider the general case of spacelike surfaces with the same mean curvature with respect to both metrics. One of our main results states that those surfaces have non-positive Gaussian curvature in R3R3. As an application of this result, jointly with a general argument on the existence of elliptic points, we present several geometric consequences for the surfaces we are considering. Finally, as any spacelike surface in L3L3 is locally a graph, our surfaces are locally determined by the solutions to the HR=HLHR=HL surface equation. Some uniqueness results for the Dirichlet problem associated to this equation are given.