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

TL;DR

This paper studies geometric properties of nonparametric $$-surfaces in $$ generated by solutions to a divergence-form PDE involving a function $g$, extending minimal surface theory to broader classes of energy densities with specific growth conditions.

Contribution

It generalizes minimal surface theory to $$-surfaces generated by solutions of a divergence PDE with $g$ of linear growth, proving closedness of a differential form and asymptotic conformal parametrization.

Findings

Proves closedness of a differential form associated with $$-surfaces.

Establishes asymptotic conformal parametrization for these surfaces.

Extends minimal surface theory to $$-surfaces with $$-elliptic energy densities.

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.