The First Eigenvalue of the Dirac Operator on Quaternionic Kähler Manifolds

Abstract

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.

The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions

“…Finding bounds for this eigenvalue has attracted much interest during the last decades. Among several estimates [Lic63,Fri80,Kir86,Kir88,KSW98,KSW99] in terms of a positive scalar curvature bound, let us mention the following estimate due to Friedrich [Fri80]. If the minimum of the scalar curvature of (M, g) is at least s > 0, then λ…”

On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a Kähler or Quaternion-Kähler structure

“…where D is the Dirac operator, and where λ 2 is one of the bounds quoted above is due to C. Bär in the general case, [Bär93], to A. Moroianu in the case of Kähler manifolds, [Mor95], [Mor99], and to W. Kramer, U. Semmelmann and G. Weingart in the case of Quaternion-Kähler manifolds, [KSW98]. The study of limiting manifolds in the Kähler and Quaternion-Kähler cases involves a special condition for spinor fields Ψ verifying (1.1), which is linked to the decomposition of the spinor space Σ into irreducible components under the action of the holonomy group.…”

Eigenvalues of the basic Dirac operator on quaternion-Kähler foliations

“…From the proof of Theorem 2.1, one gets necessarily that r = 0 and the following Equations Moreover for all X ∈ (Q), we have the quaternion-Kähler Killing equations [10,19,20]…”

“…Our approach comes from an adaptation of [10,20] to the case of Riemannian foliations where the key point is to prove that the mean curvature vanishes since the transversal Ricci curvature is strictly positive. The limiting case is characterized by the existence of quaternionKähler Killing spinors (see Section 5 for details).…”