Minimal Lagrangian surfaces in $${\mathbb {CH}^2}$$and representations of surface groups into SU(2, 1)
Minimal Lagrangian surfaces in $${\mathbb {CH}^2}$$and representations of surface groups into SU(2, 1)
复制标题
$${mathbb {CH}^2}$$中的最小拉格朗日曲面以及 SU(2, 1) 中的曲面群表示
DOI：
10.1007/s10711-012-9717-1
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发表时间：
2010
影响因子：
0.5
通讯作者：
I. McIntosh
中科院分区：
文献类型：
--
作者：
John C. Loftin;I. McIntosh
本文利用一个椭圆型微分方程，利用紧致双曲型黎曼曲面和三次全纯微分的数据，构造了一个极小Lagrange曲面。对于基本群的SU（2，1）表示，最小拉格朗日曲面是等变的.我们使用这些数据来构建Teichmuller空间（其纤维参数化立方全纯微分）上的全纯向量丛的零截面的邻域与SU（2，1）表示空间中的Fuchsian表示的邻域之间的同构。我们证明了在这个邻域中的所有表示都是复双曲拟Fuchsian的，通过为每个基本域构造一个SU（2，1）框架来实现最小拉格朗日浸入：该框架的Maurer-Cartan方程是一个双曲型方程。一个非常相似的方程，我们的管辖极小曲面在双曲3空间，我们的论文可以解释为一个模拟的理论极小曲面在准Fuchsian流形，作为第一次研究乌伦贝克。
We use an elliptic differential equation of Ţiţeica (or Toda) type to construct a minimal Lagrangian surface infrom the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for anSU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the-Fuchsian representations in theSU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using anSU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the Ţiţeica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.