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

调和映射到对称空间以及可积系统理论的应用

基本信息

批准号：
06640174
负责人：
UDAGAWA Seiichi
金额：
$ 1.15万
依托单位：
Nihon University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06640174/
关键词：
Harmonic map Symmetric space Torus Finite type Conformal map Complex Grassmannion Quaternicnic projective space Quaternionic projective space harmonic map finite type primitive map complex Grassmann manifold quaternionic projective

项目摘要

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>).

对称空间中调和二环面的构造有两种已知方法。一种方法是使用扭曲纤维的方法，产生的调和映射被认为是超极小的。另一种方法是使用可积系统理论，它是由二维线性流构造的。后一类调和映射称为有限型映射。例如，一阶紧对称空间中的任何非共形调和二环面都是有限类型的。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二环面都被有限类型的本原映射所覆盖。然后自然地产生以下问题：(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所覆盖？(2)当目标是一个秩数大于一的紧致对称空间时，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有限型本原映射所覆盖或由扭转原映射构成。对于问题(2)，G_2(C^4)中的弱共形非超极小调和二环面被有限类型的本原映射覆盖或用twistor纤维构造。在附加条件下，同型定理对G_2(C^&lt；2n&gt；)成立。