Half-space type theorems for a class of weighted minimal surfaces in $\mathbb{R}^{3}$

TL;DR

This paper proves half-space theorems for a class of height-dependent weighted minimal surfaces in three-dimensional space, showing non-existence of certain properly embedded surfaces under specific growth and curvature conditions.

Contribution

It extends classical half-space theorems to weighted minimal surfaces with height-dependent weights, including cases with quadratic growth and stochastically complete surfaces.

Findings

No proper surfaces in two transverse vertical half-spaces for quadratic weight growth

Extension of half-space results to stochastically complete weighted minimal surfaces

A version of the Hoffman-Meeks half-space theorem for surfaces with bounded principal curvatures

Abstract

We establish half-space type results for a class of height-dependent weighted minimal surfaces in $\mathbb{R}^3$, namely critical points of a weighted area functional whose weight depends on the height. When the weight has at most quadratic growth, we prove that there are no proper surfaces contained either in two transverse vertical half-spaces of $\mathbb{R}^3$ or in a half-space determined by a non-vertical plane. We show that this second result holds in a more general context, namely, for a class of stochastically complete weighted minimal surfaces. In this setup, we also prove a result for surfaces contained in regions bounded by cones. Furthermore, for stochastically complete weighted minimal surfaces satisfying restrictions on their principal curvatures, we establish a version of the classic strong half-space result due to Hoffman-Meeks.