In this paper we constructed a helicoidal surface with a light-like axis with prescribed mean curvature or Gauss curvature given by smooth function in 3-dimensional Minkowski space E 3 1 and solved an open problem left by Beneki, Kaimakamis, and Papantoniou in [

Abstract. Let M n be a closed hypersurface of constant mean curvature immersed in the unit sphere S n+1 . Denote by S the square of the length of its second fundamental form. If S < 2 √ n − 1, M is a small hypersphere in S n+1 . We also characterize all M n with S = 2 √ n − 1.

In this paper, we construct helicoidal surfaces under the cubic screw motion with prescribed mean or Gauss curvature in Minkowski 3-space E 3 1 . We also find explicitly the relation between the mean curvature and Gauss curvature of them. Furthermore, we discuss helicoidal surfaces under the cubic screw motion with H 2 = K and prove that these surfaces have equal constant principal curvatures.  2005 Elsevier Inc. All rights reserved.

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