On the spectral rigidity of Einstein-type K\"ahler manifolds

TL;DR

This paper investigates whether the eigenvalues of the p-Laplacian can determine the constant holomorphic sectional curvature of compact K"ahler manifolds, showing that for certain Einstein conditions, this spectral characterization is mostly possible.

Contribution

It extends spectral geometry results by demonstrating that under Einstein-type conditions, the holomorphic sectional curvature can be characterized by eigenvalues of the p-Laplacian for all but finitely many p and n.

Findings

Standard complex projective spaces are spectrally characterized among cohomologically Einstein K"ahler manifolds.

For p=0, complex projective spaces are characterized by the first nonzero eigenvalue among Fano K"ahler-Einstein manifolds.

The results generalize previous work and connect spectral data with geometric curvature properties.

Abstract

We are concerned in this article with a classical question in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of holomorphic sectional curvature of a complex $n$-dimensional compact K\"ahler manifold can be completely determined by the eigenvalues of its $p$-Laplacian for a \emph{single} integer $p$? We treat this question in this article under two Einstein-type conditions: cohomologically Einstein and Fano Einstein. Building on our previous work, we show that for cohomologically Einstein K\"ahler manifolds this is true for all but finitely many pairs $(p,n)$. As a consequence, the standard complex projective spaces can be characterized among cohomologically Einstein K\"ahler manifolds in terms of a single spectral set in all these cases. Moreover, in the case of $p=0$, we show that the complex projective spaces can be characterized among Fano K\"ahler-Einstein manifolds only in terms of the first nonzero eigenvalue with multiplicity, which has a similar flavor to a recent remarkable result due to Kento Fujita.