A universal lower bound for the first eigenvalue of the Dirac operator on quaternionic Kaehler manifolds

TL;DR

This paper establishes a universal lower bound for the first positive eigenvalue of the Dirac operator on compact quaternionic Kähler manifolds with positive scalar curvature, matching that of quaternionic projective space.

Contribution

It introduces a new lower bound for the Dirac operator's first eigenvalue on quaternionic Kähler manifolds, using hyperkählerian structures and twistor operators.

Findings

Lower bound equals the eigenvalue on quaternionic projective space.

The bound is derived via hyperkählerian structures on the SO(3)-bundle.

The approach involves analyzing hyperkählerian twistor operators.

Abstract

A universal lower bound for the first positive eigenvalue of the Dirac operator on a compact quaternionic Kaehler manifold M of positive scalar curvature is calculated. It is shown that it is equal to the first positive eigenvalue on the quaternionic projective space. For this, the horizontal tangent bundle on the canonical SO(3)-bundle over M is equipped with a hyperkaehlerian structure and the corresponding splitting of the horizontal spinor bundle is considered. The desired estimate is obtained by looking at hyperkaehlerian twistor operators on horizontal spinors.