The Hijazi inequalities on complete Riemannian Spinc^c manifolds

In this paper, we extend the Hijazi type inequality, involving the Energy-Momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spin$^c$ manifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality for elements of the essential spectrum. The limiting cases are also studied

Boundary value problems for noncompact boundaries of Spinᶜ manifolds and spectral estimates

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. B\"ar and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi-Montiel-Zhang, involving the mean curvature of the boundary, for the spectrum of the Dirac operator on the noncompact boundary of a Spin$^c$ manifold. The limiting case is then studied and examples are then given.Comment: Accepted in Proceedings of the London Mathematical Societ

Spinorial Characterization of CR Structures, I

We characterize certain CR structures of arbitrary codimension (different from 3, 4 and 5) on Riemannian Spin$^c$ manifolds by the existence of a Spin$^c$ structure carrying a strictly partially pure spinor field. Furthermore, we study the geometry of Riemannian Spin$^c$ manifolds carrying a strictly partially pure spinor which satisfies the generalized Killing equation in prescribed directions.Comment: This paper has been withdrawn by the author. All results have been changed because we extended them to all co-dimensions. We changed completely the pape

Hypersurfaces of Spinc^c manifolds and Lawson type correspondence

Simply connected 3-dimensional homogeneous manifolds $E(\kappa, \tau)$, with 4-dimensional isometry group, have a canonical Spin$^c$ structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $E(\kappa, \tau)$. As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in $E(\kappa, \tau)$. Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spin$^c$ spinors.Comment: to appear in Annals of Global Analysis and Geometry (AGAG

Eigenvalue Estimates of the spinc{\rm spin}^c Dirac Operator and Harmonic Forms on K\"ahler-Einstein Manifolds

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact K\"ahler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of K\"ahlerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact K\"ahler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing spinor is zero

Complex and Lagrangian surfaces of the complex projective plane via K\"ahlerian Killing Spinc^c spinors

The complex projective space $\mathbb C P^2$ of complex dimension $2$ has a Spin$^c$ structure carrying K\"ahlerian Killing spinors. The restriction of one of these K\"ahlerian Killing spinors to a surface $M^2$ characterizes the isometric immersion of $M^2$ into $\mathbb C P^2$ if the immersion is either Lagrangian or complex.Comment: 18 page

Complex Generalized Killing Spinors on Riemannian Spinc^c manifolds

In this paper, we extend the study of generalized Killing spinors on Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin$^c$ Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing that the dimension of the manifold is greater or equal to 4. Moreover, we classify Riemannian Spin$^c$ manifolds carrying imaginary and imaginary generalized Killing spinors.Comment: 15 page