Some geometric properties of nonparametric $\mu$-surfaces in $\mathbb{R}^3$

Abstract

Smooth solutions of the equation \[ \rm{div}\, \Bigg\{ \frac{g'\big(|\nabla u|\big)}{|\nabla u|} \nabla u \Bigg\} = 0 \] are considered generating nonparametric $\mu$-surfaces in $\mathbb{R}^3$, whenever $g$ is a function of linear growth satisfying in addition \[ \int_0^\infty s g''(s) d s < \infty \, . \] Particular examples are $\mu$-elliptic energy densities $g$ with exponent $\mu > 2$ (see [1]) 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 $\hat{N} \wedge d X$. As a corollary we find an asymptotic conformal parametrization generated by this differential form.