论文标题
一个不稳定标准，用于支撑区域固定的表面，具有单数曲线，以下曲线$ 3 $ - 空间表格
An instability criterion for volume-preserving area-stationary surfaces with singular curves in sub-Riemannian $3$-space forms
论文作者
论文摘要
我们研究稳定的表面，即固定体积变化的区域的二阶最小值，以尺寸为$ 3 $的亚riemannian空间形式。我们证明了稳定不平等，并提供了足够的条件，以确保具有非空的单数曲线的不稳定面积$ c^2 $表面。结合以前的结果，这允许描述Heisenberg Group $ \ Mathbb {H}^1 $的任何完整，定向，嵌入式和稳定的$ C^2 $表面$σ$和Sub-Riemannian Sphere $ \ Mathbb {S}^3 $ constant countance curvature curvature curvature $ 1 $ $。在$ \ mathbb {h}^1 $中，我们得出结论，$σ$是欧几里得飞机，pansu球或与双曲线抛物面$ t = xy $的一致。在$ \ mathbb {s}^3 $中，我们推断出$σ$是[28]中发现的Pansu球形表面之一。结果，此类领域是$ \ Mathbb {s}^3 $中的sub-riemannian等等问题的唯一$ c^2 $解决方案。
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 $Σ$ 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 $Σ$ is a Euclidean plane, a Pansu sphere or congruent to the hyperbolic paraboloid $t=xy$. In $\mathbb{S}^3$ we deduce that $Σ$ is one of the Pansu spherical surfaces discovered in [28]. As a consequence, such spheres are the unique $C^2$ solutions to the sub-Riemannian isoperimetric problem in $\mathbb{S}^3$.
