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.     

Metrics with harmonic spinors

We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS ⁿ ,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM ₁#M ₂ with certain metrics is close to the union of the spectra ofM ₁ and ofM ₂.Partially supported by SFB 256 and by the GADGET program of the EU     

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.     

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.

     

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.

     

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.     

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.     

Normal families of harmonic functions

A condition on the majorant (depending on a single coordinate) of a family of functions harmonic in a sphere is determined whose satisfaction guarantees normality of the family.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 547–552, May, 1968.]     

Nehari-type families of harmonic mappings

We introduce affine and linearly invariant families of locally injective harmonic mappings of the unit disk $\mathbb{D}$. We derive sharp distortion theorems for the Jacobian that are used to establish a uniform modulus of continuity for the quasiconformal mappings in each class. Finally, we find a converse of a recent theorem of Chen and Ponnusamy characterizing when the image $f(\mathbb{D})$ under a quasiconformal harmonic univalent mapping is a John domain.     

Representation formulas for integrated semigroups and sine families

Summary The purpose of this paper is to obtain representation formulas for an integrated semigroup and a sine family of linear operators in terms of its generator.     

Quaternionic vector coherent states and the supersymmetric harmonic oscillator

The quaternionic vector coherent states are realized as coherent states of the supersymmetric harmonic oscillator with broken symmetry in analogy with the standard canonical coherent states of the ordinary harmonic oscillator. We study the nonclassical properties of the oscillator, such as the photon number distribution and signal-to-quantum-noise ratio in terms of these states and discuss the squeezing properties and the temporal stability of the coherent states. We obtain the orthogonal polynomials associated with the quaternionic vector coherent states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 80–98, October, 2006.     

Bohr radius for subordinating families of analytic functions and bounded harmonic mappings

The class consisting of analytic functions f   in the unit disk satisfying f+αzf^′+γz²f^″$f+\alpha z{f}^{\prime }+\gamma z^2{f}^{″}$ subordinated to some function h is considered. The Bohr radius for this class is obtained when h is respectively convex or starlike. The Bohr radius for analytic functions mapping the unit disk into a concave-wedge domain as well as for bounded harmonic mappings are also established.     

Four families of summation formulas involving generalized harmonic numbers

In terms of Abel’s transformation on difference operators, we establish four families of summation formulas involving generalized harmonic numbers. They include several known and numerous new harmonic number identities as special cases.     

Representation of harmonic functions as potentials and the Cauchy problem

In this paper, we propose an explicit formula for the reconstruction of a harmonic function in the domain from its known values and from the values of its normal derivative on part of the boundary, i.e., we give an explicit continuation and a regularization formula of the solution of the Cauchy problem for the Laplace equation.