Spinc geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

Abstract

Abstract. From the existence of parallel spinor fields on CalabiYau, 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.

Spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

“…In the three cases, there exists a canonical frame {e 1 , e 2 , e 3 } such that the Christoffel symbols are given by (12). From these relations, we can see that the curvature operator R, acting on 2-forms, defined by…”

“…The spinorial approach allows to solve naturally some problems of geometry of submanifolds. For instance, some simple proofs of the Alexandrov theorem in the Euclidean space were given ( [13]) or in the hyperbolic space ( [11]) and new results about Einstein manifolds or Langrangian submanifolds of Kähler manifolds ( [12]). …”

Boundary value problems for noncompact boundaries of Spincmanifolds and spectral estimates

“…, where K M is the canonical bundle of M [13,22,21,24]. Let α be the Kähler form defined by the complex structure J, i.e.…”

“…The auxiliary line bundle L = (K M ) −1 has a canonical holomorphic connection induced from the Levi-Civita connection whose curvature form is given by iΩ = iρ, where ρ is the Ricci 2-form given by ρ(X, Y ) = Ric(X, JY ). Here Ric denotes the Ricci tensor of M. For any other Spin c structure on M 2m , the spinorial bundle can be written as [13,21]:…”

“…(ii) Every almost complex manifold (M 2m=n+1 , g, J) of complex dimension m has a canonical Spin c structure. In fact, the complexified cotangent bundle T * M ⊗ C = Λ 1,0 M ⊕ Λ 0,1 M decomposes into the ±i-eigenbundles of the complex linear extension of the complex symbol, we have on Σ (see [15,22,23]) that…”