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.

一段时间以来，人们已经知道类型为\（A_2 ^ {（2）} \）的复仿射Kac-Moody代数存在5种基本不同的实型，并且可以将其中的4种实型与某些类别的“可积分曲面”，例如\（\ mathbb {CP} ^ 2 \）和\（\ mathbb {CH} ^ 2 \）中的最小拉格朗日曲面，以及\（{\ mathbb { R}} ^ 3 \）。在本文中，我们考虑了不定复双曲双空间\（\ mathbb {CH} ^ {2} _1 \）中的类时间最小拉格朗日曲面。我们证明了这类表面对应于第五种实数形式。此外，对于\（\ mathbb {CH} ^ {2} _1 \）中的每个时空拉格朗日曲面 我们将自然高斯图定义到某些齐次空间中，并证明了Ruh-Vilms型定理，根据这些高斯图的调和性来刻画所有时态拉格朗日曲面之间的时态最小拉格朗日曲面。