Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

TL;DR

This paper classifies timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space, linking them to a specific real form of an affine Kac-Moody algebra and characterizing them via harmonic Gauss maps.

Contribution

It identifies the class of timelike minimal Lagrangian surfaces with the fifth real form of a complex affine Kac-Moody algebra and establishes a Ruh-Vilms type theorem for their characterization.

Findings

Timelike minimal Lagrangian surfaces correspond to the fifth real form of the algebra.

Gauss maps into homogeneous spaces are harmonic for these surfaces.

A Ruh-Vilms type theorem characterizes these surfaces via harmonic Gauss maps.

Abstract

It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb {CH}^2$, as well as definite and indefinite affine spheres in $\mathbb R^3$. In this paper we consider the class of timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space $\mathbb{CH}^2_1$. We show that this class of surfaces corresponds to the fifth real form. Moreover, for each timelike Lagrangian surface in $\mathbb {CH}^2_1$ we define natural Gauss maps into certain homogeneous spaces and prove a Ruh-Vilms type theorem, characterizing timelike minimal Lagrangian surfaces among all timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps.