A non-existence theorem for pluriharmonic maps of finite energy (Q1089652)

A map \(f: M\to N\) between Kähler manifolds is said to be pluriharmonic if \(\nabla_{1,0}{\bar \partial}f=0\), where \(\nabla_{1,0}\) is the covariant differentiation of type (1,0) relative to \(\nabla\) induced on \(T^*M\otimes f^*TN\). Then: holomorphic \(\Rightarrow\) pluriharmonic \(\Rightarrow\) harmonic. The author's main result asserts that for M complete, \(n=\dim M\geq 2\), carrying a special potential function \(\Phi\) any pluriharmonic map of finite energy of M into any Kähler manifold N is constant. Here \(\Phi\) is a real valued \(C^ 2\) function such that \(| \text{grad} \Phi | \leq 1\), \(\Phi\) is hyper \(n-1\) convex on M and strongly n-1 convex at some point of M, i.e. the sum of any \(n-1\) eigenvalues of the Hermitian matrix \((\Phi_{i\bar j})\) is \(\geq 0\) on M and \(>0\) at some point of M. Specific examples for such M include Kähler Cartan-Hadamard manifolds, bounded domains carrying a complete Kähler metric on any Stein manifolds etc. The result is also applicable to harmonic maps f if the range has nonpositive Riemannian curvature in the sense of Nakano since in this case f is pluriharmonic.