Weierstrass Representation For Surfaces Of Prescribed Mean Curvature In The 3-Dimensional Heisenberg Group

In this paper, we define the Gauss map of surfaces in the 3-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature. Furthermore, we obtain a second order partial differential equation for the Gauss map and show that this equation is the complete integrabihty condition of the above obtained representation.This paper is made up of five parts:In section one, we introduce the backgroud of the Weierstrass representation and the relavant research progress, what's more, we state our main results.In section two, we give some basic formulas in a surface theory and define the Gauss map of surfaces in Nil₃.In section three, we obtain a Weierstrass representation formula for surfaces of prescribed mean curvature.In section four, we show that the Gauss map of an arbitrary surface in Nil₃ satisfies a second order differential equation which is a complete integrabihty condition for the above Weierstrass representation.In section five, we apply the Weierstrass representation formula to give some examples.