Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and SL (3, C) -quasi-Fuchsian representations

In this paper we introduce complex minimal Lagrangian surfaces in the bi-complex hyperbolic space and study their relation with representations in SL (3, C). Our theory generalizes at the same time minimal Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres in R³, and Bers embeddings in the holomorphic space form CP¹ CP¹. If these surfaces are equivariant under representations in SL (3, C), our approach generalizes the study of almost R-Fuchsian representations in SU (2, 1), Hitchin representations in SL (3, R), and quasi-Fuchsian representations in SL (2, C). Moreover, we give a parameterization of SL (3, C) -quasi-Fuchsian representations by an open set in the product of two copies of the bundle of holomorphic cubic differentials over the Teichm\"uller space of S, from which we deduce that this space of representations is endowed with a bi-complex structure. In the process, we introduce bi-complex Higgs bundles as a new tool for studying representations into semisimple complex Lie groups.