Almost harmonic spinors

We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.     

Surgery and harmonic spinors

Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.     

Principal series representations and harmonic spinors

Let G be a real reductive Lie group and G/H a reductive homogeneous space. We consider Kostant's cubic Dirac operator D on G/H twisted with a finite-dimensional representation of H. Under the assumption that G and H have the same complex rank, we construct a nonzero intertwining operator from principal series representations of G into the kernel of D. The Langlands parameters of these principal series are described explicitly. In particular, we obtain an explicit integral formula for certain solutions of the cubic Dirac equation D=0 on G/H.     

Non--trivial harmonic spinors on generic algebraic surfaces

We show that there are simply connected spin algebraic surfaces for which all complex structures in certain components of the moduli space admit more harmonic spinors than predicted by the index theorem (or Riemann--Roch). The dimension of the space of harmonic spinors can exceed the absolute value of the index by an arbitrarily large number.

     

Representation theoretic harmonic spinors for coherent families

Coherent continuation π ₂ of a representation π ₁ of a semisimple Lie algebra arises by tensoring π ₁ with a finite dimensional representation F and projecting it to the eigenspace of a particular infinitesimal character. Some relations exist between the spaces of harmonic spinors (involving Kostant’s cubic Dirac operator and the usual Dirac operator) with coefficients in the three modules. For the usual Dirac operator we illustrate with the example of cohomological representations by using their construction as generalized Enright-Varadarajan modules. In [9] we considered only discrete series, which arises as generalized Enright-Varadarajan modules in the particular case when the parabolic subalgebra is a Borel subalgebra.     

Partially harmonic spinors and representations of reductive Lie groups

Let G be a reductive Lie group subject to some minor technical restrictions. The Plancherel Theorem for G uses several series of unitary representation classes, one series for each conjugacy class of Cartan subgroups of G. Given a Cartan subgroup H ? G, we construct a G-homogeneous family X → Y of oriented riemannian symmetric spaces, some G-homogeneous bundles , and some Hilbert spaces of partially harmonic spinors with values in . Then G acts on by a unitary representation π_μ,σ^±. We then show that these π_μ,σ^± realize the series of representation classes of G associated to the conjugacy class of H.     

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 remark on the space of metrics having nontrivial harmonic spinors

Let M be a closed spin manifold of dimension n ≡ 3 mod 4. We give a simple proof of the fact that the space of metrics on M with invertible Dirac operator is either empty or it has infinitely many path components.     

On spinors

For a 2^n-dimensional complex Hermitian vector space S, we prove that any unitary basis of S can be explained as an augmented spinor structure on S. By using this explanation, a SpinC（2n）- action on S is equivalent to an action on a subset of augmented spinor structures. The latter action is a little easy to be understood, and is shown in the last part of this paper. Such kind of understanding could be of use to the discussions of Hermitian manifolds and spin manifolds, especially could help to find connections and elliptical operators.     

Extrinsic Killing spinors

 Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non-negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary isometric to a round sphere, is necessarily a flat disc. Received: 2 February 2002; in final form: 1 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 53C27, 53C40, 53C80, 58G25 The authors would like to thank Lars Andersson for helpful discussions and for bringing to our knowledge the information regarding Remark 4. We are also grateful to the referee for pointing out that Corollary 5 and Corollary 6 are only valid when the boundary is at least 2-dimensional. Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967     

Symmetric spinors ink-dimensions

Summary In his paper, explicit formulae are given for any irreducible spinor, in any number of dimensions, which is symmetric in its suffixes. The method employed in determining these formulae is immediately applicable to the task of obtaining explicit forms for spinors which are unsymmetric in their suffixes. Entrata in Redazione il 22 maggio 1971.     

Skew Killing spinors

We study the existence of a skew Killing spinor on 2- and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.     

On non-simply connected manifolds with non-trivial parallel spinors

We examine the possibilities of the full holonomy groups of locally irreducible but not necessarily complete Riemannian spin manifolds admitting a non-trivial parallel spinor and discuss some applications of this classification.partially supported by NSERC Grant No. OPG0009421     

On the stability of Riemannian manifold with parallel spinors

Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results. Dedicated to Jeff Cheeger for his sixtieth birthday     

Codazzi spinors and globally hyperbolic manifolds with special holonomy

In this paper, we describe the structure of Riemannian manifolds with a special kind of Codazzi spinors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy surface for any weakly irreducible holonomy representation with parallel spinors, t.m. with a holonomy group , where is trivial or a product of groups SU(k), Sp(l), G ₂ or Spin (7).