The geometry of maximal representations of surface groups into SO(2,n)

TL;DR

This paper explores the geometric and dynamical aspects of maximal surface group representations into SO(2,n), revealing their structure, spectral properties, and associated minimal surfaces using advanced mathematical tools.

Contribution

It introduces new results on the geometric structures, spectral bounds, and minimal surface preservation for maximal representations into SO(2,n), extending prior theorems in the field.

Findings

Maximal representations are holonomies of specific geometric structures.

Their length spectrum exceeds that of certain Fuchsian representations.

They preserve a unique minimal surface in the symmetric space.

Abstract

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.