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

Matteo Galli

Vol. 271 (2014), No. 1, 143–157

DOI: 10.2140/pjm.2014.271.143

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 $E\left(1,1\right)$, equipped with its subriemannian structure. We construct examples of area-stationary surfaces that are not foliated by subriemannian 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 subriemannian geodesics.

Keywords

subriemannian geometry, area-stationary surfaces, stable surfaces, pseudohermitian manifolds, Sol geometry

Mathematical Subject Classification 2010

Primary: 49Q05, 49Q20, 53C17

Milestones

Received: 27 May 2013

Revised: 9 August 2013

Accepted: 16 August 2013

Published: 10 September 2014