We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension 3. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary \(C^2\) surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable \(C^2\) surface \(\Sigma \) in the Heisenberg group \({\mathbb {H}}^1\) and the sub-Riemannian sphere \({\mathbb {S}}^3\) of constant curvature 1. In \({\mathbb {H}}^1\) we conclude that \(\Sigma \) is a Euclidean plane, a Pansu sphere or congruent to the hyperbolic paraboloid \(t=xy\). In \({\mathbb {S}}^3\) we deduce that \(\Sigma \) is one of the Pansu spherical surfaces discovered in Hurtado and Rosales (Math Ann 340(3):675–708, 2008). As a consequence, such spheres are the unique \(C^2\) solutions to the sub-Riemannian isoperimetric problem in \({\mathbb {S}}^3\).

我们研究尺寸为3的次黎曼空间形式的稳定表面，即固定体积变化的面积的二阶最小值。我们证明了稳定性不等式，并提供了充分的条件来确保体积保持面积固定的不稳定性\（C ^ 2 \）曲面具有一组非空的奇异曲线。与先前的结果相结合，这允许描述任何完整，定向，嵌入式和稳定\（C ^ 2 \）表面\（\西格玛\）海森堡组中\（{\ mathbb {H}} ^ 1 \）和恒定曲率1的次黎曼球面\（{\ mathbb {S}} ^ 3 \）在\（{\ mathbb {H}} ^ 1 \）中，我们得出\（\ Sigma \）是一个欧几里得平面，一个Pansu球面或与双曲线抛物面\（t = xy \）相等。在\（{\ mathbb {S}} ^ 3 \）中，我们推论\（\ Sigma \）是在Hurtado和Rosales中发现的Pansu球形表面之一（Math Ann 340（3）：675–708，2008）。结果，这样的球体是\（{\ mathbb {S}} ^ 3 \）中次黎曼等渗问题的唯一\（C ^ 2 \）解。