New development of harmonic maps

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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.