#### Vol. 271, No. 1, 2014

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

### Matteo Galli

Vol. 271 (2014), No. 1, 143–157
##### 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
##### Authors
 Matteo Galli Departamento de Geometría y Topología Universidad de Granada 18071 Granada Spain