Affine Translation Surfaces In Euclidean 3-space

The surfaces theory is an important part in differential geometry. In Minkowski 3-space and Euclidean 3-space, the translation surface is very special and has been studied by many experts and scholars. In Minkowski 3-space, according to the translation direction, the translation surfaces can be considered as six types. In Euclidean 3-space, there is one type.It is well-known that the surfaces whose Gaussian curvature and mean curvature satisfy some relationships are Weingarten surfaces. The properties of the surfaces can be decided by Gaussian curvature and mean curvature, so it is very important to study the Weingarten surfaces. In Euclidean 3-space, many people have studied the Weingarten translation surfaces, but not Weingarten affine translation surfaces.The affine translation surfaces that we discuss in this paper are the translation surfaces under affine coordinate system.In this paper we study affine translation surfaces of different types in Euclidean 3-space and give some classifications of such surfaces whose mean curvature and Gauss curvature are constants. Meanwhile the concrete theorems about different kinds of affine translation surfaces are given in this paper. It is well-known that the minimal translation surface in Euclidean 3-space is Scherk surface. Scherk surface is a kind of classical surface. The main work of this paper is that we find a new type of minimal surface which is different from Scherk surface. Minimal surface has been the subject that many geometers are keen on and an important part of differential geometry. It has deep connections with partial differential equations, the complex function, topology, differential geometry. So the finding of this minimal surface enriches the theory of minimal surface and is very important in the future study.