Spectral rigidity of complex projective spaces, revisited

TL;DR

This paper investigates whether spectral data of Laplacian eigenvalues on complex projective spaces uniquely determine their geometry, affirming this for most cases and clarifying previous gaps in the literature.

Contribution

It proves spectral rigidity for complex projective spaces in most cases, resolving longstanding questions and addressing gaps in prior research.

Findings

Spectral rigidity holds for all but two cases in the specified range.

Confirmed that Laplacian eigenvalues determine the geometry of complex projective spaces.

Clarified previous misconceptions related to volume estimates of Fano Kähler-Einstein manifolds.

Abstract

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact K\"{a}hler manifold are the same as those of $\mathbb{C}P^n$ equipped with the Fubini-Study metric, then whether or not this K\"{a}hler manifold is holomorphically isometric to $\mathbb{C}P^n$. For every positive even number $p$, we affirmatively solve this problem in all dimensions $n$ with at most two possible exceptions. We also clarify in this paper some gaps in previous literature concerned with this question, among which one is related to the volume estimate of Fano K\"{a}hler-Einstein manifolds.