Lower bounds for the eigenvalues of the Dirac operator on Spinc manifolds

Abstract

In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on Spin c manifolds without boundary. The limiting case is then studied and an example is given.

A note on twisted Dirac operators on closed surfaces

“…We can think of the Dirac operator associated to a Spin c structure as the twisted Dirac operator acting on spinors which are twisted by a line bundle. For more details on Spin c structures see [19, Appendix D] and [13,23] for eigenvalue estimates of the Spin c Dirac operator. For this reason the eigenvalue estimate (1.2) can also be interpreted as an eigenvalue estimate of the Dirac operator associated to any Spin c structure and (1.4) as an estimate on the nodal set of a spinor that is in the kernel of the Spin c Dirac operator.…”

THE ENERGY-MOMENTUM TENSOR ON LOW DIMENSIONAL Spinc MANIFOLDS

“…In this section, we begin with some preliminaries concerning Spin c structures and the Dirac operator. Details can be found in [18], [20], [7], [23] and [24].…”

“…Next, we will give another proof of the Bär-type inequality (1.3) for the eigenvalues of any Spin c Dirac operator. The following theorem was proved by the second author in [23] using conformal deformation of the spinorial Levi-Civita connection.…”

“…Additionally, any compact surface or any product of a compact surface with R has a Spin c structure carrying particular spinors. In the same spirit as in [14], when using a suitable conformal change, the second author [23] established a Bär-type inequality for the eigenvalues of the Dirac operator on a compact surface endowed with any Spin c structure. In fact, any eigenvalue λ of the Dirac operator satisfies…”

The Hijazi Inequalities on Complete Riemannian Spinc Manifolds

“…Here X · ψ denotes the Clifford multiplication and ∇ the spinorial Levi-Civita connection [9,16]. In [20], it is shown that on a compact Riemannian Spin c manifold any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies the Hijazi type inequality [15] involving the Energy-Momentum tensor and the scalar curvature:…”