Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group

TL;DR

This paper characterizes when graphical strips in the Heisenberg group are area-minimizing, proving that such entire ruled graphs are vertical planes and exploring boundary curves with multiple minimal fillings.

Contribution

It provides a necessary and sufficient condition for area-minimizing graphical strips in the Heisenberg group, advancing understanding of minimal surfaces in sub-Riemannian geometry.

Findings

Characterization of area-minimizing graphical strips in the Heisenberg group

Proof that entire ruled minimal graphs are vertical planes

Identification of boundary curves with uncountably many minimal fillings

Abstract

In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group $\mathbb{H}$ to be area-minimizing in the slab $\{-1<x<1\}$. We show that our condition is necessary by introducing a family of deformations of graphical strips based on varying a vertical curve. We show that it is sufficient by showing that strips satisfying the condition have monotone epigraphs. We use this condition to show that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that admits uncountably many fillings by area-minimizing surfaces.