HELICOIDAL SURFACES OF THE THIRD FUNDAMENTAL FORM IN MINKOWSKI 3-SPACE

Abstract

Abstract. We study helicoidal surfaces with the non-degenerate thirdfundamental form in Minkowski 3-space. In particular, we mainly focuson the study of helicoidal surfaces with light-like axis in Minkowski 3-space. As a result, we classify helicoidal surfaces satisfying an equationin terms of the position vector field and the Laplace operator with respectto the third fundamental form on the surface. 1. IntroductionWe study a (pseudo-)Riemannian manifold as a submanifold of a (pseudo-)Euclidean space via an isometric immersion by Nash’s Theorem. Let x :M → E 3 be an isometric immersion of a connected surface M in a Euclidean3-space E 3 . Denote by ∆ the Laplacian with respect to the induced metric onM. Takahashi ([11]) proved that minimal surfaces and spheres are the onlysurfaces in E 3 satisfying the condition ∆x = λx, λ ∈ R. As a generalization ofTakahashi’s Theorem, Garay ([4]) classified the hypersurfaces whose coordinatefunctions in E m are eigenfunctions of their Laplacian, that is, the hypersurfacesin E