Minimal translation surface in H2 X R

Abstract

In mathematics and in physics, a minimal surface is a surface minimizes its area while achieving certain conditions on board. In elemental di erentiel geometry, a minimal surface is a closed and bounded surface of a real Euclidean space of dimension 3 with regular board minimizing the total area with xed contour. In 1744, Leonhard Euler posed and solved the rst minimal surface problem : nding between all surfaces passing through two parallel circles, the one with the smallest surface. In particular, as the study of minimum surfaces, L.Euler found that the only minimum surfaces of revolution are planes and catenoids. In 1760, Lagrange generalised Euler's results for calculating variations for integrals to one variable in the case of two variables. He sought to solve the following problem : given a q 3 closed curve of E , to determine a minimum area having this curve as a boundary such a q surface is called a minimum area. In 1776, Meusnier showed that the di erential equation obtained by Lagrange being equivalent to a condition on the mean curvature : an area is minimal if and only if its mean q curvature at any point is zero . q 3 3 3 2 2 We have eight homogeneous spaces of dimension 3 : E ,H , S , S ×R,H ×R, SL (2,R) ,N il 3 2 and Sol . In particular, our study will be space H × R. 3 In this brief we have made it possible to obtain classi cation results concerning the 2 minimum translation areas of two properly prolonged types in the H × R space. From D. W. Yoon's article , we will address the following information : 2 2 Let H be represented by the upper half-plane model {(x, y) ∈ R |y > 0} equipped 2 2 2 2 with the metric g = (dx + dy )⧸y . The space H , with the group structure derived by H the composition of proper a ne maps, is a Lie group and the metric g is left invariant. H 2 Therefore the Riemannian product space H ×R is a Lie group with respect to the operetion (x, y, z) ∗ (¯x, ¯ y, ¯ z) = (¯xy + x, y¯ y, z + ¯ z) and the left invariant product metric 2 2 dx + dy 2 g = + dz .