Eigenvalue estimate of the basic Dirac operator on a Kähler foliation of even complex codimension

Abstract

In this paper, we prove that on a Kähler spin foliation of codimension q = 2 n ( n = even ) any eigenvalue λ of the basic Dirac operator D b satisfies the inequality λ 2 ⩾ n 4 ( n − 1 ) inf M { σ ∇ + | κ | 2 } , where κ is the mean curvature form and σ ∇ is the transversal scalar curvature of F . In the limiting case, the foliation F is minimal.