Related papers: Eigenvalue estimates for the Dirac operator and ha…

We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds…

In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic…

We solve for spectrum, obtain explicitly and study group properties of eigenfunctions of Dirac operator on the Riemann sphere $S^2$. The eigenvalues $\lambda$ are nonzero integers. The eigenfunctions are two-component spinors that belong to…

On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first…

We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the…

On a n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric, a spin structure and a chirality operator, we study some properties of a spin conformal invariant defined from the first eigenvalue of…

A closed spin K\"ahler manifold of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator is characterized by holomorphic spinors. It is shown that on any spin K\"ahler-Einstein manifold each holomorphic…

We prove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particluar we give a lower bound for the first eigenvalue of the Dirac operator for non-trivial spin structures. It…

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a…

We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such…

In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex…

We extend to the eigenvalues of the hypersurface Spin$^c$ Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Futhermore, we prove a correspondence between the existence of a Spin$^c$ Killing spinor…

Let us fix a conformal class $[g_0]$ and a spin structure $\sigma$ on a compact manifold $M$. For any $g\in [g_0]$, let $\lambda^+_1(g)$ be the smallest positive eigenvalue of the Dirac operator $D$ on $(M,g,\sigma)$. In a previous paper we…

We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on Sasakian and on 3-dimensional manifolds and partially classify those satisfying…

It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove that over…

Given a closed Riemannian Spin manifold $(M,g)$ of dimension greater or equal than four, we consider a generalized conformally invariant equation involving the Dirac operator with a non-linearity of convolution type. We show that the…

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number…

In K\"ahler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which…

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_\xi(S)$. The definition of $L^2_\xi(S)$ involves the choice of a…

We consider the Dirac operator on compact quaternionic Kaehler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.