The classification of complete stable area-stationary surfaces in the Heisenberg group $\mathbb{H}^1$

TL;DR

This paper classifies all complete, stable, area-stationary surfaces in the Heisenberg group, showing they are either planes or hyperbolic paraboloids, thus advancing understanding of geometric structures in sub-Riemannian spaces.

Contribution

It provides a complete classification of stable area-stationary surfaces in the Heisenberg group, identifying them explicitly as planes or hyperbolic paraboloids.

Findings

Complete stable surfaces are either planes or hyperbolic paraboloids.

Classification applies to $C^2$ smooth, orientable, connected surfaces.

Results deepen understanding of sub-Riemannian geometry in $ extbf{H}^1$.

Abstract

We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.