We consider the Dirac operator on compact quaternionic Kähler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.

In [KSW97a] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler 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 introduced in [KSW97b] and show that a nontrivial solution defines a parallel spinor on the associated hyperkähler manifold.

No abstract

No abstract

No abstract