Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations

TL;DR

This paper introduces complex minimal Lagrangian surfaces in bi-complex hyperbolic space, linking them to $ ext{SL}(3, ext{C})$ representations, and develops a new bi-complex Higgs bundle framework for studying such representations.

Contribution

It generalizes existing theories of minimal Lagrangian surfaces and quasi-Fuchsian representations, and introduces bi-complex Higgs bundles as a novel tool for complex Lie group representations.

Findings

Parameterization of $ ext{SL}(3, ext{C})$-quasi-Fuchsian representations

Establishment of a bi-complex structure on the representation space

Introduction of bi-complex Higgs bundles for complex Lie groups

Abstract

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.