On the holomorphicity of proper harmonic maps between unit balls with the Bergman metrics
On the holomorphicity of proper harmonic maps between unit balls with the Bergman metrics
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DOI：
10.1007/s002080050015
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发表时间：
2000-02
影响因子：
1.4
通讯作者：
S. Li;Lei Ni
中科院分区：
文献类型：
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作者：
S. Li;Lei Ni
设Mm和Nn为两个Kähler流形，分别具有Kähler度量h= hij dzidzj和g= gαβdwαdwβ。让u: M→N是一个地图从M N M和N都紧凑,在他著名的强刚性定理的证明Kahler导管紧凑,Siu (S1)证明任何谐波必须全纯映射u或antiholomorphic,假设N强烈负曲率的Siu du一度的秩是大于或等于4个(最后一个条件不包括复杂的维度的情况下一个定理时显然是错误的)。证明的关键是Siu的∂∂-Bochner公式：
Let Mm and Nn be two Kähler manifolds with Kähler metrics h= hij dzidzj and g= gαβdwαdwβ, respectively. Let u: M→ N be a map from M to N. When both M and N are compact, in his proof of the celebrated strong rigidity theorem for compact Kähler manifolds, Siu [S1] proved that any harmonic map u must be holomorphic or antiholomorphic, under the assumption that N has strongly negative curvature in the sense of Siu and the rank of du at one point is greater than or equal to four (the last condition excludes the case of complex dimension one when the theorem is obviously false). The key of the proof is Siu’s∂∂-Bochner formula: