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

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 ${\mathbb{CH}}_1^2$. We show that this class of surfaces corresponds to the fifth real form. Moreover, for each timelike Lagrangian surface in ${\mathbb{CH}}_1^2$ 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.

Typological errors have been fixed

References in corpus (2)

• On Indefinite Special Lagrangian Submanifolds in Indefinite Complex Euclidean Spaces
• Survey on real forms of the complex $A_2^{(2)}$-Toda equation and surface theory