On the classification of complete area-stationary and stable surfaces in the sub-Riemannian Sol manifold

TL;DR

This paper classifies complete area-stationary and stable surfaces in the sub-Riemannian Sol manifold, providing new examples and stability results, advancing understanding of geometric structures in this non-Euclidean setting.

Contribution

It introduces new examples of area-stationary surfaces not foliated by geodesics and proves stability of certain foliated surfaces in the sub-Riemannian Sol manifold.

Findings

Existence of infinitely many area-stationary surfaces with singular curves

Construction of non-geodesic foliated area-stationary surfaces

Stability results for surfaces foliated by sub-Riemannian geodesics

Abstract

We study the classification of area-stationary and stable $C^2$ regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We also prove that there exist an infinite number of $C^2$ area-stationary surfaces with a singular curve. Finally we show the stability of $C^2$ area-stationary surfaces foliated by sub-Riemannian geodesics.