A characterization of complex projective space up to biholomorphic isometry
The authors characterize complex projective space up to biholomorphic isometry by the existence of a solution of a system of second order equations. Their main theorem is the following. Theorem. Let \(M\) be a complex manifold of dimension \(\geq 2,g\) a complete Kähler metric on \(M\). Then \(M\) is biholomorphically isometric to complex projective space \({\mathbb P}^n\) with the Fubini-Study metric if and only if there is a nonconstant real-valued function \(u\in C^2(M)\) such that on \(\{p\in M\mid \text{grad } u(p)\neq 0\}\), \(\text{Hess } u = -uId + {1\over 2} (u-1)(Id-\pi)\), where \(\pi = \pi_{\text{grad}_hu} + \pi_{\text{grad}_{\bar h}u}\). Here, the operator \(\pi\) is the projection onto the complex subspace of \(T^{\mathbb C}_PM\) spanned by \(\text{grad}_hu\) and \(\text{grad}_{\bar h}u\), or, equivalently, by grad \(u\) and \(J\) grad \(u\) (where \(J\) is the complex structure operator).
scientific article; zbMATH DE number 1326041