Eigenvalue value estimates and stability of positive quaternion-K\"ahler manifolds

TL;DR

This paper investigates the stability of positive quaternion-Kähler manifolds, providing eigenvalue estimates and characterizing deformations using Laplace eigenfunctions and symmetric tensors.

Contribution

It offers new eigenvalue bounds for the Hodge-Laplacian and characterizes destabilizing directions in the stability analysis.

Findings

Sharp lower bound for the first non-zero eigenvalue on Sym^2 E

Description of infinitesimal Einstein deformations in terms of eigenfunctions

Improved eigenvalue estimates for the Hodge-Laplacian on 2-forms

Abstract

In this article we study the stability problem for positive quaternion-K\"ahler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of symmetric 2-tensors. We also give improved eigenvalue estimates for the Hodge-Laplacian on 2-forms. On the parallel subbundle Sym^2 E of the 2-form bundle we prove a sharp lower bound for the first non-zero eigenvalue.