论文标题
细长的曲线，极限集和球形CR均匀化
Slim curves, limit sets and spherical CR uniformisations
论文作者
论文摘要
我们在这里考虑$ 3 $ -SPHERE $ \ MATHBF S^3 $被视为复杂双曲机平面的无穷大的边界$ \ MATHBF {H}^2 _ {\ MATHBF C} $。它配备了接触结构和两类特殊曲线。第一个$ \ mathbf r $ -circles是无穷大的边界，完全是真正的完全测量子空间，并且与触点分布相切。其次，$ \ mathbf c $ -circles，它们是复杂的完全测量子空间的边界，并且横向到触点分布。 我们定义了一个定量概念，称为Slimness，该概念在$ \ mathbf s^3 $中的连续路径在多大程度上接近是$ \ Mathbf r $ circle。我们分析了$ \ Mathbf c $ -circles的$ \ mathbf r $ -circle的补充的经典叶子。接下来，我们考虑$ \ mathbf r $ -circle变成细长的曲线的这种情况的变形。我们将这些概念应用于特定情况，即纤细曲线是$ \ mathrm {pu}（2,1）$的准英式亚组的极限集。结果，我们描述了某些cus $ 3 $ manifolds的一类球形CR均匀化。
We consider here the $3$-sphere $\mathbf S^3$ seen as the boundary at infinity of the complex hyperbolic plane $\mathbf{H}^2_{\mathbf C}$. It comes equipped with a contact structure and two classes of special curves. First $\mathbf R$-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, $\mathbf C$-circles, which are boundaries of complex totally geodesic subspaces and are transverse to the contact distribution. We define a quantitative notion, called slimness, that measures to what extent a continuous path in the sphere $\mathbf S^3$ is near to be an $\mathbf R$-circle. We analyze the classical foliation of the complement of an $\mathbf R$-circle by arcs of $\mathbf C$-circles. Next, we consider deformations of this situation where the $\mathbf R$-circle becomes a slim curve. We apply these concepts to the particular case where the slim curve is the limit set of a quasi-Fuchsian subgroup of $\mathrm{PU}(2,1)$. As a consequence, we describe a class of spherical CR uniformizations of certain cusped $3$-manifolds.
