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

Abstract

AbstractSmooth solutions of the equation $$\begin{aligned} {\text {div}}\, \Bigg \{ \frac{g'\big (|\nabla u|\big )}{|\nabla u|} \nabla u \Bigg \} = 0 \end{aligned}$$ div { g ′ ( | ∇ u | ) | ∇ u | ∇ u } = 0 are considered generating nonparametric $$\mu $$ μ -surfaces in $${\mathbb {R}}^3$$ R 3 , whenever g is a function of linear growth satisfying in addition $$\begin{aligned} \int _0^\infty s g''(s) \mathrm{d}s < \infty . \end{aligned}$$ ∫ 0 ∞ s g ′ ′ ( s ) d s < ∞ . Particular examples are $$\mu $$ μ -elliptic energy densities g with exponent $$\mu > 2$$ μ > 2 (see Bildhauer 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 $${\hat{N}}\wedge \mathrm{d}X$$ N ^ ∧ d X . As a corollary we find an asymptotic conformal parametrization generated by this differential form.