论文标题
叶状高原问题和表面亚组的渐近计数
Foliated Plateau problems and asymptotic counting of surface subgroups
论文作者
论文摘要
在[17]中，Labourie启动了$ k $ - 曲面空间的动力学特性的研究，也就是说，在$ 3 $ 3 $维的歧管中，合适的完整浸入式浸入式表面持续不断的外部曲率表面，当他以较高的二维类模拟时，当Ambient流形的情况下，他将其作为较高的二维类似物表现出来。在本文中，在Calegari-Marques的最新工作[5]之后，我们研究了表面亚组的渐近计数，从$ k $ surfaces的面积来看。我们确定一个下限，并在实现这种界限时证明了刚性。我们的工作在两个关键方面与[5]的作品不同。首先，我们与所有准富奇西亚的亚组合作，而不是渐近的紫红色亚组。其次，随着[5]中的刚性证明在当前情况下破裂，我们需要采取不同的方法。遵循Labourie在[19]中概述的想法之后，我们通过解决cartan的普通高原问题（Hadamard歧管）来证明刚性。为此，我们基于Labourie的$ k $ -surface Dynamics的理论，并提出了许多新的结构，猜想和问题。
In [17], Labourie initiated the study of the dynamical properties of the space of $k$-surfaces, that is, suitably complete immersed surfaces of constant extrinsic curvature in $3$-dimensional manifolds, which he presented as a higher-dimensional analogue of the geodesic flow when the ambient manifold is negatively curved. In this paper, following the recent work [5] of Calegari--Marques--Neves, we study the asymptotic counting of surface subgroups in terms of areas of $k$-surfaces. We determine a lower bound, and we prove rigidity when this bound is achieved. Our work differs from that of [5] in two key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to merely asymptotically Fuchsian ones. Secondly, as the proof of rigidity in [5] breaks down in the present case, we require a different approach. Following ideas outlined by Labourie in [19], we prove rigidity by solving a general foliated Plateau problem in Cartan--Hadamard manifolds. To this end, we build on Labourie's theory of $k$-surface dynamics, and propose a number of new constructions, conjectures and questions.
