  Spacelike surfaces in the Lorentz-Minkowski space L^3 can be endowed with two
different Riemannian metrics, the metric inherited from L^3 and the one induced
by the Euclidean metric of R^3. 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
R^3. 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 L^3 is
locally a graph over a domain of the plane x_3=0, our surfaces are locally
determined by the solutions to the H_R=H_L surface equation. Some uniqueness
results for the Dirichlet problem associated to this equation are given.
