A characterization of complex space forms via Laplace operators

Abstract

Inspired by the work of Lu and Tian (Duke Math J 125(2):351–387, 2004), in this paper we address the problem of studying those Kähler manifolds satisfying the \(\Delta\)-property, i.e. such that on a neighborhood of each of its points the kth power of the Kähler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer k (see below for its definition). We prove two results: (1) if a Kähler manifold satisfies the \(\Delta\)-property then its curvature tensor is parallel; (2) if an Hermitian symmetric space of classical type satisfies the \(\Delta\)-property then it is a complex space form (namely it has constant holomorphic sectional curvature). In view of these results we believe that if a Kähler manifold satisfies the \(\Delta\)-property then it is a complex space form.