Spin$^c$ geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

Oussama Hijazi (1) , S. Montiel , F. Urbano

1 IECN - Institut Élie Cartan de Nancy

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers. When the ambient manifold is Kähler-Einstein with positive 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 spin$^c$ structures. 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.

Mots clés: Spin$^c$ geometry; Dirac operator; Lagrangian submanifolds; Hodge Laplacian

Domaines: Géométrie différentielle [math.DG]

Oussama Hijazi, S. Montiel, F. Urbano. Spin$^c$ geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds. Mathematische Zeitschrift, 2006, 253, Number 4, pp.821--853. ⟨hal-00098171⟩