On the $\Delta$-property for complex space forms

TL;DR

This paper proves the conjecture that K"ahler manifolds satisfying the $ riangle$-property are necessarily complex space forms, advancing understanding of geometric structures with specific Laplacian properties.

Contribution

It provides a proof confirming that the $ riangle$-property characterizes complex space forms among K"ahler manifolds.

Findings

Confirmed the conjecture linking $ riangle$-property to complex space forms.

Established that the $k$-th power of the K"ahler Laplacian is polynomial in the Euclidean Laplacian.

Enhanced understanding of the geometric implications of the $ riangle$-property.

Abstract

Inspired by the work of Z. Lu and G. Tian [8], A. Loi, F. Salis and F. Zuddas address in [5] the problem of studying those K\"ahler manifolds satisfying the $\Delta$-property, i.e. such that on a neighborhood of each of its points the $k$-th power of the K\"ahler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer $k$. In particular, they conjectured that if a K\"ahler manifold satisfies the $\Delta$-property then it is a complex space form. This paper is dedicated to the proof of the validity of this conjecture.