We study uniformization problems for compact manifolds that arise as quotients of domains in complex flag varieties by images of Anosov homomorphisms. We focus on Anosov homomorphisms with “small” limit sets, as measured by the Riemannian Hausdorff codimension in the flag variety. Under such a codimension hypothesis, we show that all first-order deformations of complex structure on the associated compact complex manifolds are realized by deformations of the Anosov homomorphism. With some mild additional hypotheses we show that the character variety maps locally homeomorphically to the (generalized) Teichmüller space of the manifold. In particular this provides a local analogue of the Bers Simultaneous Uniformization Theorem in the setting of Anosov homomorphisms to higher-rank complex semisimple Lie groups.

我们通过 Anosov 同态的图像研究作为复杂标志变体中域的商的紧凑流形的均匀化问题。我们关注具有“小”极限集的 Anosov 同态，这是由标志变体中的 Riemannian Hausdorff codimension 测量的。在这样的余维假设下，我们证明了相关联的紧凑复流形上复结构的所有一阶变形都是通过 Anosov 同态的变形来实现的。通过一些温和的附加假设，我们表明字符多样性局部同胚映射到流形的（广义）Teichmüller 空间。特别是，这在 Anosov 同态到更高阶复数半单李群的设置中提供了 Bers 同时均匀化定理的局部模拟。