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 \(\mathbb {CH}^{2}_1\). 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.