Harmonic maps into symmetric spaces and applications of the theory of integrable systems

对称空间中调和双环面的构造有两种方法。一种方法是使用扭量纤维化的方法，并且所得到的调和映射被称为是超极小的。另一种方法是利用可积系统理论，从二维线性流出发构造。后一种调和映射称为有限型。例如，秩为1的紧致对称空间中的任何非共形调和双环面都是有限型的。Burstall证明了球面或复射影空间中的任何弱共形非超极小调和双环面都被一个有限型本原映射覆盖。这样就自然而然地产生了以下问题：（1）四元数射影空间中的弱共形非超极小调和二环面是否被有限型本原映射覆盖？(2)如果目标是秩大于1的紧致对称空间，情况又如何呢？我们对问题（1）的结果是：HP ^3中的任何弱共形非超极小调和二环面都被一个有限型本原映射覆盖或由扭量纤维化构造。对于问题（2），G_2（C^4）中的弱共形非超极小调和二环面被一个有限型本原映射覆盖或利用扭量纤维化构造。在附加条件下，对G_2（C^），同样的定理也成立<2n>.

The construction of harmonic two-tori in symmetric spaces are known in two ways. One way is a method using twistor fibration and the resulting harmonic map is said to be superminimal. Another way is a method using the theory of integrable system and it is constructed from two-dimensional linear flows. The latter harmonic map is said to be finite type. For example, any non-conformal harmonic two-tori in compact symmetric spaces of rank one is of finite type. Burstall proved any weakly conformal non-superminimal harmonic two-tori in a sphere or a complex projective space is covered by a primitive map of finite type. Then the following problems naturally arises :(1) Is weakly conformal non-superminimal hatmonic two-tori in quaternionic projective space covered by a primitive map of finite type?(2) How about the case where the target is a compact symmetric space of rank greater than one?Our result for the problem (1) is : Any weakly conformal non-superminimal harmonic two-tori in HP^3 is covered by a primitive map of finite type or constructed by using twistor fibration. For the problem (2), weakly conformal non-superminimal harmonic two-tori in G_2(C^4) is covered by a primitive map of finite type or constructed by using twistor fibration. Under the additional condition, the same type theorem holds for G_2(C^<2n>).