Eigenvalues of Type r of the Basic Dirac Operator on Kähler Foliations

Abstract

Abstract. In this paper, we prove that on a Kahler spin foliatoin of codimension q = 2n,any eigenvalue λ of type r (r ∈ {1,··· ,[ n+12 ]}) of the basic Dirac operator D b satisfies theinequality λ 2 ≥ r4r−2 inf M σ ∇ , where σ ∇ is the transversal scalar curvature of F. 1. IntroductionOn a K¨ahler spin foliation (M,F) of codimension q = 2n, any eigenvalue λ ofthe basic Dirac operator D b satisfies(1.1) λ 2 ≥n+14ninf M K σ if n is odd [6,7],n4(n−1)inf M K σ if n is even [4],where K σ = σ ∇ + |κ| 2 with the transversal scalar curvature σ ∇ and the meancurvature form κ of F. In the limiting cases, F is minimal. For the point foliation,see [9,10]. Since the limiting cases of (1.1) are minimal, the inequalities (1.1) yieldthe following:(1.2) λ 2 ≥n+14ninf M σ ∇ if n is odd,n4(n−1)inf M σ ∇ if n is even.In this paper, we give an estimate of the eigenvalues λ of type r of the basic Diracoperator D b on a K¨ahler spin foliation. Recently, G. Habib and K. Richardson [5]proved that the spectrum of the basic Dirac operator does not change with respectto a change of bundle-like metric. And the existence of a bundle-like metric suchthat δ