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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2019
- DOI:
- 10.48550/arXiv.1909.04818
- arXiv:
- arXiv:1909.04818
- Bibcode:
- 2019arXiv190904818D
- Keywords:
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- Mathematics - Differential Geometry;
- Primary 53A10;
- 53B30;
- 58D10;
- Secondary 53C42
- E-Print:
- Typological errors have been fixed