Some Geometric Properties of Nonparametric $$\mu $$-Surfaces in $$\pmb {{\mathbb {R}}}^3$$

Abstract

Smooth solutions of the equation div g

|∇u|

|∇u| ∇u

= 0 are considered generating nonparametric μ-surfaces in R3, whenever g is a function of linear growth satisfying in addition ∞ 0 sg (s)ds < ∞. Particular examples are μ-elliptic energy densities g with exponent μ > 2 (see Bild hauer and Fuchs in Rend Mat Appl 22(7):249–274, 2003) and the minimal surfaces belong to the class of 3-surfaces. Generalizing the minimal surface case we prove the closedness of a suitable differential form Nˆ ∧dX. As a corollary we find an asymptotic conformal parametrization generated by this differential form.