Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-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 $\mathrm{SL}(3,\mathbb{C})$. Our theory generalizes at the same time minimal Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres in $\mathbb{R}^3$, and Bers embeddings in the holomorphic space form $\mathbb{CP}^1 \times \mathbb{CP}^1 \setminus \Delta$. If these surfaces are equivariant under representations in $\mathrm{SL}(3,\mathbb{C})$, our approach generalizes the study of almost $\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$, Hitchin representations in $\mathrm{SL}(3,\mathbb{R})$, and quasi-Fuchsian representations in $\mathrm{SL}(2,\mathbb{C})$. Moreover, we give a parameterization of $\mathrm{SL}(3,\mathbb{C})$-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.