The First Eigenvalue of the Dirac Operator on Quaternionic Kaehler Manifolds

TL;DR

This paper characterizes quaternionic projective spaces as the unique manifolds where the first eigenvalue of the Dirac operator attains its lower bound, using quaternionic Killing equations and hyperkähler geometry.

Contribution

It proves the uniqueness of quaternionic projective spaces in the spectral limit case for the Dirac operator on quaternionic Kähler manifolds.

Findings

Quaternionic projective spaces attain the lower bound of the Dirac spectrum.

Solutions to the quaternionic Killing equation correspond to parallel spinors.

The result links spectral properties to geometric structures of manifolds.

Abstract

In a previous paper we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler manifolds. In the present article we show that the only manifolds in the limit case, i.e. the only manifolds where the lower bound is attained as an eigenvalue, are the quaternionic projective spaces. We use the equivalent formulation in terms of the quaternionic Killing equation and show that a nontrivial solution defines a parallel spinor on the associated hyperkaehler manifold.