Area-stationary and stable surfaces of class $C^1$ in the sub-Riemannian Heisenberg group ${\mathbb H}^1$

TL;DR

This paper characterizes $C^1$ surfaces in the Heisenberg group that are area-stationary and stable, showing they are foliated by horizontal lines or are vertical planes, with implications for minimal surface classification.

Contribution

It proves that complete, stable, $C^1$ area-stationary surfaces in ${ m H}^1$ are vertical planes, extending Bernstein-type results to less regular surfaces.

Findings

Regular parts of area-stationary surfaces are foliated by horizontal lines.

Stable, complete, $C^1$ surfaces without singular points are vertical planes.

Complete, locally area-minimizing $C^1$ graphs are vertical planes.

Abstract

We consider surfaces of class $C^1$ in the $3$-dimensional sub-Riemannian Heisenberg group ${\mathbb H}^1$. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In case the surface is complete and oriented, without singular points, and stable, i.e., a second order minimum of perimeter, we prove that the surface must be a vertical plane. This implies the following Bernstein type result: a complete locally area-minimizing intrinsic graph of a $C^1$ function in ${\mathbb H}^1$ is a vertical plane.