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

Smooth solutions of the equation $\rm{div}\, \Bigg\{ \frac{g'\big(|\nabla u|\big)}{|\nabla u|} \nabla u \Bigg\} = 0$ are considered generating nonparametric $μ$-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 $μ$-elliptic energy densities $g$ with exponent $μ> 2$ (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.

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