复$A_2^{(2)}$ -Toda方程的实型及曲面理论综述
Survey on real forms of the complex $A_2^{(2)}$-Toda equation and surface theory
摘要：
描述从曲面到约化李群的对称空间的调和映射的经典结果表明，带有一个附加参数的Maurer-Cartan形式，即所谓的回路参数，对于回路参数的所有值都是可积的。事实上，对于约化李群上的$k$ -对称空间，同样的结果也成立。本文将证明对于$A_2^{(2)}$的一个环群的五种不同实型中的每一种都存在一个曲面类，对于该曲面类，某个框架对于环参数的所有值都是可积的当且仅当它属于其中一个曲面类，即$\mathbb {CP}^2$中的极小拉格朗日曲面，$\mathbb {CH}^2$中的极小拉格朗日曲面，$\mathbb {CH}^2_1$中的类时极小拉格朗日曲面，分别是$\mathbb R^3$中的真定仿射球和$\mathbb R^3$中的真不定仿射球。
英文摘要：
The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for $k$-symmetric spaces over reductive Lie groups. In this survey we will show that to each of the five different types of real forms for a loop group of $A_2^{(2)}$ there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in $\mathbb {CP}^2$, minimal Lagrangian surfaces in $\mathbb {CH}^2$, timelike minimal Lagrangian surfaces in $\mathbb {CH}^2_1$, proper definite affine spheres in $\mathbb R^3$ and proper indefinite affine spheres in $\mathbb R^3$, respectively.
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