Helicoidal Surfaces And Pseudo-umbilical Surfaces In Minkowski 3-space

As the space-time model of Einstein's Relativity thoery, Minkowski space catches more and more mathematicians' attention. Compared with Euclidean space, Minkowski space is a fire-new field. Because of the difference of its metric from Euclidean space's, many essential concepts in Minkowski space change a lot. This makes the conclusions of some problems in Minkowski space much different. We always consider the properties possessed by Minkowski space but not by Euclidean space because these properties reflect the essence of Minkowski space more thoroughly. According to this idea we study two types of surfaces, i.e., helicoidal surfaces and pseudo-umbilical surfaces, in Minkowski 3-space.A helicoidal surface in Minkowski 3-space is defined as the orbit of a plane curve under a screw motion. Depending on the axis being spacelike, timelike or null, there are three types of screw motions, the third case, i.e., cubic screw motion, is most particular because it is not a rotation together with a translation along the null axis, but a rotation together with a translation along a null curve with non-zero curvature.In chapter 3 we distinguish helicoidal surfaces in Minkowski 3-space into five cases and discuss two cases under the cubic screw motion intensively. We choose a pseudo-orthonormal basis as the coordinate frame and parametrize the mean curvature and the Gauss curvature in more simple form. In fact, the mean curvature and Gauss curvature of any helicoidal surface are independent on rotation parameter. Based on this property, by solving a series ODEs we construct a two-parametric family of helicoidal surfaces under cubic screw motion with prescribed mean curvature or Gauss curvature and solve an open problem left by Beneki. Thus, combining this work with Beneki's work, we prove that there exist any type of helicoidal surface with mean curvature or Gauss curvature being any smooth function. Moreover, we discuss some geometrical properties about helicoidal surfaces in Minkowski 3-space.A pseudo-umbilical surfaces in Minkowski 3-space is defined as a surface with the mean curvature H and Gauss curvature K satisfying H~2 = K (Locally, surfaces with this property in Euclidean 3-space can be classified into plans and two spheres). In chapter4, by choosing two families of null coordinate curves, we prove that any pseudo-umbilical surface in Minkowski 3-space is a null scroll. Moreover, we classify pseudo-umbilical surfaces in Minkowski 3-space into the following cases:? Spacelike case: Hyperbolic space H2, spacelike plan.? Timelike case: Null scroll- Timelike umbilical surface: de Sitter 2-space S2, timelike plan.- Generalized umbilical surface: B-scroll.- Null scroll with shape operator (x - b)2(b ^ constant).Obviously, the last two types of timelike cases have no counterparts in Euclidean space. Especilly, we prove that any pseudo-umbilical helicoidal surface in Minkowski 3-space is a ï¿¡?-scroll.