Eigenvalue estimates for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form

On a compact Riemannian manifold MM with a transverse spin foliation FF of codimension q≥3q≥3, if MM admits a non-trivial basic harmonic 1-form ωω, then any eigenvalue λλ of the basic Dirac operator satisfies the inequality λ2≥q−14(q−2)infM(σ∇+|κ|2), where σ∇σ∇ is the transversal scalar curvature and κκ is the mean curvature form of FF. In the limiting case, FF is minimal and ωω is parallel.