On a conjecture about the Gauss map of complete spacelike surfaces with constant mean curvature in the Lorentz-Minkowski space (Q1875913)

By a conjecture proposed by do Carmo in 1981 the Gauss map image of every compact surface in \(\mathbb R^3\) with nonzero constant mean curvature contains a maximal circle of the sphere. Up to now the conjecture seems to be open except assuming additional properties. Especially, the conjecture is true in case of helicoidal surfaces and as a special case for the well known Delaunay surfaces (Seaman, 1984). The present paper deals with a corresponding conjecture in Lorentz-Minkowski 3-space \(L^3\): Given a complete space-like surface in \(L^3\) with nonzero constant mean curvature, its Gauss map image contains an arbitrary maximal geodesic of the (one sheeted) hyperboloid in \(L^3\). The conjecture is true in case of space-like rotational surfaces.