论文标题
在浸入SL（2，C）和几何后果中的表面上
On immersions of surfaces into SL(2,C) and geometric consequences
论文作者
论文摘要
我们将平滑流形的完全浸入恒定截面曲率-1的全态riemannian空间形式的完全真实的浸入研究中。我们介绍了第一和第二基本形式的概念，我们证明它们满足了经典的高斯 - 古斯齐方程的类似版本，相反，高斯 - 科达兹方程的解决方案是某些均等地图的沉浸数据。这项研究有一些有趣的几何后果： 1）它提供了一种形式主义，可以研究将表面浸入SL（2，c）和H^3的大地测量空间中的形式主义； 2）它概括了浸入非零曲率空间形式的经典理论，从而导致了在H^n，ads^n，ds^n和s^n之间过渡的模型； 3）我们证明，浸入式数据的全态家族对应于浸入式浸入式家族，提供了一种有效的方法来构建全体形态图中的（n，c） - 特征品的品种。特别是，我们将指出复杂滑坡的全体形态性的更简单证明。 4）我们看到，在某些假设下，恒定曲率-1的表面上的复杂指标（即其复杂的切线束的复杂双线性形式）对应于具有相同固体的射击表面对。在这种结构上应用BERS双均匀定理，我们证明了表面上复杂指标的均匀定理。
We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a similar version of the classic Gauss-Codazzi equations, and conversely that solutions of Gauss-Codazzi equations are immersion data of some equivariant map. This study has some interesting geometric consequences: 1) it provides a formalism to study immersions of surfaces into SL(2,C) and into the space of geodesics of H^3; 2) it generalizes the classical theory of immersions into non-zero curvature space forms, leading to a model for the transitioning of hypersurfaces among H^n, AdS^n, dS^n and S^n; 3) we prove that a holomorphic family of immersion data corresponds to a holomorphic family of immersions, providing an effective way to construct holomorphic maps into the SO(n,C)-character variety. In particular we will point out a simpler proof of the holomorphicity of the complex landslide; 4) we see how, under certain hypothesis, complex metrics on a surface (i.e. complex bilinear forms of its complexified tangent bundle) of constant curvature -1 correspond to pairs of projective surfaces with the same holonomy. Applying Bers Double Uniformization Theorem to this construction we prove a Uniformization Theorem for complex metrics on a surface.
