SPINc GEOMETRY OF KÄHLER MANIFOLDS AND THE HODGE LAPLACIAN ON MINIMAL LAGRANGIAN SUBMANIFOLDS

Abstract. From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the ex-istence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the sub-manifolds are compact, we obtain sharp estimates on their Betti numbers. When the ambient manifold is Kähler-Einstein with pos-itive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spinc struc-tures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. Applications on the Morse index of minimal Lagrangian submanifolds are obtained. 1