Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let F be a Kähler spin foliation of codimension q=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75-90] that the eigenvalue λ of the basic Dirac operator Db satisfies the inequality , where σ∇ is the transversal scalar curvature of F. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.     

A Sobolev-like inequality for the Dirac operator

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 nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

A vanishing result for strictly p-convex domains

In view of Andreotti and Grauert (Bull Soc Math France 90:193–259, 1962) vanishing theorem for \(q\) -complete domains in \(\mathbb C ^{n}\) , we reprove a vanishing result by Sha (Invent Math 83(3):437–447, 1986), and Wu (Indiana Univ Math J 36(3):525–548, 1987), for the de Rham cohomology of strictly \(p\) -convex domains in \(\mathbb R ^n\) in the sense of Harvey and Lawson (The foundations of \(p\) -convexity and \(p\) -plurisubharmonicity in riemannian geometry. arXiv:1111.3895v1 [math.DG]). Our proof uses the \({L}^2\) -techniques developed by Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Publishing Co, Amsterdam 1990), and Andreotti and Vesentini (Inst Hautes Études Sci Publ Math 25:81–130, 1965).     

A matroid generalization of a result of Dirac

This paper generalizes a theorem of Dirac for graphs by proving that ifM is a 3-connected matroid, then, for all pairs {a,b} of distinct elements ofM and all cocircuitsC ^* ofM, there is a circuit that contains {a,b} and meetsC ^*. It is also shown that, although the converse of this result fails, the specified condition can be used to characterize 3-connected matroids.The author's research was partially supported by a grant from the National Security Agency.     

Half-inverse problem for the Dirac operator

Given the spectrum of the Dirac operator, together with the potential on the half-interval and one boundary condition, this paper provides reconstruction of the potential on the whole interval, and proves the existence conditions of the solution.     

A spectral estimate for the Dirac operator on Riemannian flows

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 both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.     

Invariant syzygies for the Hermitian Dirac operator

This paper is devoted to the algebraic analysis of the system of differential equations described by the Hermitian Dirac operators, which are two linear first order operators invariant with respect to the action of the unitary group. In the one variable case, we show that it is possible to give explicit formulae for all the maps of the resolution associated to the system. Moreover, we compute the minimal generators for the first syzygies also in the case of the Hermitian system in several vector variables. Finally, we study the removability of compact singularities. We also show a major difference with the orthogonal case: in the odd dimensional case it is possible to perform a reduction of the system which does not affect the behavior of the free resolution, while this is not always true for the case of even dimension. A. Damiano is a postdoctoral fellow at the Eduard Čech Center and is supported by the relative grants. D. Eelbode is a postdoctoral fellow supported by the F.W.O. Vlaanderen (Belgium).     

The index of the pseudo-Riemannian Dirac operator as a transversally elliptic operator

Let (M,r) be a closed, space- and time-orientable, pseudo-Riemannian spin manifold and-let G be a compact group of orientation-preserving isometries on (M,r). If there are no isotropic directions transversal to the orbits of G, then the Dirac operator on (M,r) is transversally elliptic. In this paper we calculate its index.     

Generic vanishing for harmonic spinors of twisted Dirac operators

In this paper we address the problem of generic vanishing for (negative) harmonic spinors of Dirac operators coupled with variable metric connections.

     

A Maurey type result for operator spaces

The little Grothendieck theorem for Banach spaces says that every bounded linear operator between C(K) and ?₂ is 2-summing. However, it is shown in [M. Junge, Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’, Invent. Math. 161 (2) (2005) 225-286] that the operator space analogue fails. Not every cb-map is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem: every cb-map is (q,cb)-summing for any q>2 and hence admits a factorization ‖v(x)‖?c(q)‖vcb_‖‖axbq_‖ with a,b in the unit ball of the Schatten class S2q.     

Extrinsic estimates for eigenvalues of the Dirac operator

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.     

Prescribing eigenvalues of the Dirac operator

In this note we show that every compact spin manifold of dimension ≥3 can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.     

Spectrum of the periodic Dirac operator

The absolute continuity of the spectrum for the periodic Dirac operator