Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Abstract

Let F be a Kähler spin foliation of codimension q = 2 n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75–90] that the eigenvalue λ of the basic Dirac operator D b satisfies the inequality λ 2 ⩾ n + 1 4 n inf M { σ ∇ + | κ | 2 } , where σ ∇ is the transversal scalar curvature of F . In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.