A characterization of complex space forms via Laplace operators

TL;DR

This paper investigates extit{K} manifolds with the extDelta-property, showing they have parallel curvature tensors and that classical Hermitian symmetric spaces with this property are complex space forms, suggesting a characterization of such manifolds.

Contribution

It proves that extit{K} manifolds with the extDelta-property have parallel curvature tensors and classifies classical Hermitian symmetric spaces with this property as complex space forms.

Findings

extit{K} manifolds with the extDelta-property have parallel curvature tensors.

Classical Hermitian symmetric spaces with the extDelta-property are complex space forms.

Conjecture that complete, simply-connected extit{K} manifolds with the extDelta-property are complex space forms.

Abstract

Inspired by the work of Z. Lu and G. Tian \cite{lutian}, in this paper we address the problem of studying those \K\ manifolds satisfying the $\Delta$-property, i.e. such that on a neighborhood of each of its points the $k$-th power of the \K 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 \K\ 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 complete and simply-connected \K\ manifold satisfies the $\Delta$-property then it is a complex space form.