Associated Minimal Surfaces In Euclidean 3-space

A minimal surface is a surface with vanishing mean curvature. The theory of minimal surface is a content-rich part, and it is one of the most studied targets in differential geometry. Associated minimal surfaces form a one-parameter group of minimal surfaces. We only need to study one of them, then we can know the properties of the group. The objects of this paper are minimal translation surfaces and minimal affine translation surfaces in Euclidean 3-space.In chapter 1, we introduce the history and the development of minimal surfaces, the research methods and the current state of minimal surfaces.In chapter 2, we give the equation of minimal surfaces, translation surfaces, affine translation surfaces, associated minimal surfaces, the formula of Weierstrass-Enneper and its application.Chapter 3 and 4 are the main work of the paper. In chapter 3, we prove the theory that, in Euclidean 3-space, Scherk surface is the only minimal surface of minimal translation surfaces. Then we give the representation of its associated minimal surfaces. Finally we obtain the conjugate minimal surface of Scherk surface that is shx·shy=-sinz. When θ equals to π, its associated minimal surface becomes z=lncosx/cosy. In chapter 4, we give the representation of minimal affine translation surfaces and the method how to obtain its associated minimal surfaces.At last, we sum up the whole paper.