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 Kahler manifolds satisfying the $$\Delta$$ -property, i.e. such that on a neighborhood of each of its points the kth power of the Kahler 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 Kahler 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 Kahler manifold satisfies the $$\Delta$$ -property then it is a complex space form.