On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space

Abstract

Let RV1 be the (n+l)-dimensional Minkowski space, that is, Rn+l with the Lorentz metric < ,)―(dx1)2+ ・・・+(dxn)2―(dxn+1)z. It has been known that in R +1 hyperplanes are the only complete space-like hypersurfaces whose mean curvatures are zero. This Bernstein type theorem was proposed by Calabi, and solved by him [3] (for n^4) and by Cheng and Yau [5] (for all n) (see alsoIshihara [10] or Nishikawa [14]). On the other hand, for complete space-like hypersurfaces of nonzero constant mean curvature in Rnx+ there are many nonlinear examples constructed by Treibergs [18], Hano and Nomizu [7], Isbihara and Hara [11] and others. In his recent paper, Palmer [17] discussed the Gauss map of a complete space-like hypersurface of constant mean curvature in 1??+1 and showed a condition for the hypersurface to be a hyperplane. This is a result analogous to the one obtained by Hoffman, Osserman and Schoen [9], who proved that the normals to a complete surface of constant mean curvature in the 3-dimensional Euclidean space Es cannot lie in a closed hemisphere of S*, unless the surface is a plane or a right circular cylinder. Note that a right circular cylinder is the simplest example of a complete non-umbilical surface of constant mean curvature in Es. In R+1 the simplest example of a complete non-umbilical space-like hypersurface of constant mean curvature is given by the following: