New development of harmonic maps

1986年，引入了双调和映射的概念，它是调和映射的扩展。我们提出了双谐波映射的类比——双杨-米尔斯场的新概念，并给出了它的孤立现象。也就是说，具有正Ricci曲率的紧流形上某些平方积分范数的bi-Yang-Mills场必须是Yang-Mills场。我们证明了具有有界平方积分范数的双调和映射在目标空间具有非正曲率时必须是调和的。我们对目标空间为紧李群或紧对称空间的所有双调和映射进行了分类。

In 1986, the concept of the bi-harmonic map which is an extension of harmonic maps was introduced. We raised the new notion of the bi-Yang-Mills field, which is an analog of the bi-harmonic map, and showed its isolation phenomena. That is, bi-Yang-Mills fields with some square-integral norm over compact manifolds with positive Ricci curvature must be Yang-Mills fields. We showed bi-harmonic maps which have a bounded square-integral norm must be harmonic if the target space has non-positive curvature. We classified all the bi-harmonic maps for cases where the target space is a compact Lie group or compact symmetric spaces.