Minimal Translation Surfaces Research Articles

Minimal translation surfaces in hyperbolic space

In the half-space model of hyperbolic space, that is, $${\mathbb{R}^3_{+}=\{(x,y,z)\in\mathbb{R}^3;z > 0\}}$$ with the hyperbolic metric, a translation surface is a surface that writes as z = f(x) + g(y) or y = f(x) + g(z), where f and g are smooth functions. We prove that the only minimal translation surfaces (zero mean curvature in all points) are totally geodesic planes.

Affine Translation Surfaces with Constant Gaussian Curvature

Abstract. We study a–ne translation surfaces in R 3 and get a complete classiflcation ofsuch surfaces with constant Gauss-Kronecker curvature. 1. IntroductionA surface in E 3 is called a translation surface if it is obtained as a graph of a func-tion F ( x;y ) = p ( x )+ q ( y ), where p ( x ) and q ( y ) are difierentiable functions. It’s wellknown that a minimal translation surface in the Euclidean space E 3 must be a planeor a Scherk surface, which is the graph of the function F ( x;y ) = ln(cos x= cos y ),the only doubly periodic minimal translation surface.In this note, we study nondegenerate translation surfaces in a–ne space R 3 .This class of surfaces has been studied previously by many geometers. F. Manhart[3] classifled all the nondegenerate a–ne minimal translation surfaces in a–ne spaceR 3 . Further treatments are due to H. F. Sun [5], who classifled the nondegeneratea–ne translation surface with nonzero constant mean curvature in R 3 . Later on,Sun and Chen extended this into the case of hypersurfaces [6]. On the other hand,Binder [1] classifled locally symmetric a–ne translation surfaces in R