Clifford algebra approach to pointwise convergence of Fourier series on spheres Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

Clifford algebra approach to pointwise convergence of Fourier series on spheres

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter’s Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini’s type pointwise convergence theorem are proved. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Exotic characteristic classes of quaternionic bundles

For aC ^∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kähler manifold. These results are the complex analoga of those given for the higher order Maslov classes in [V2].     

Dynamics on Teichmüller spaces and self-covering of Riemann surfaces

A non-injective holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmüller space. We investigate the dynamics of such self-embeddings by applying our structure theorem of self-covering of Riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the Teichmüller space. We also extend our observation to quasiregular self-covers of Riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self-cover.     

Complex structures in the Nash-Moser category

Working in the Nash-Moser category, it is shown that the harmonic and holomorphic differentials and the Weierstrass points on a closed Riemann surface depend smoothly on the complex structure. It is also shown that the space of complex structures on any compact surface forms a principal bundle over the Teichmüller space and hence that the uniformization maps of the closed disk and the sphere depend smoothly on the complex structure.     

A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions

We study a decomposition of a holomorphic vector bundle with connection which need not be endowed with any metrics, which is a generalization of an orthogonal decomposition of a Hermitian holomorphic vector bundle. We first derive several results on the induced connections, the second fundamental forms of subbundles and curvature forms of the connections. We next apply these results to a complex affine immersion. Especially, we give elementary self-contained proofs of the fundamental theorems for a complex affine immersion to a complex affine space.     

On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions

It has been found recently that there exists a theory of functions with quaternionic values and in two real variables, which is determined by a Cauchy–Riemann‐type operator with quaternionic variable coefficients, and that is intimately related to the so‐called Mathieu equations. In this work, it is all explained as well as some basic facts of the arising quaternionic function theory. We establish analogues of the basic integral formulas of complex analysis such as Borel–Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. This theory turns out to be in the same relation with the Schrödinger operator with special potential as the usual holomorphic functions in one complex variable, or quaternionic hyperholomorphic functions, or functions of Clifford analysis, are with the corresponding Laplace operator. Moreover, it is similar to that of α‐hyperholomorphic functions and the Helmholtz operator. Copyright © 2012 John Wiley & Sons, Ltd.     

Vector Bundles on Certain Non-algebraic Surfaces

It is well-known that every holomorphic vector bundle is filtrable on a projective algebraic     

The Prym and Ganning Vector Bundles over the Teichmueller Space

We introduce and study the Prym vector bundle P of holomorphic Prym differentials and the Ganning cohomology bundle G over the Teichmueller space of compact Riemann surfaces of genus g2 and over the Torelli space of genus g2. We construct a basis of holomorphic Prym differentials on a variable compact Riemann surface which depends on the moduli of the compact Riemann surface and on the essential characters. From these bundles we compose an exact sequence of holomorphic vector bundles over the product of the Teichmueller space of genus g and a special domain in the complex manifold C ^2g/Z ^2g.     

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

The space of harmonic Prym differentials on a variable compact Riemann surface

The harmonic Prym differentials and their period classes play an important role in the modern theory of functions on compact Riemann surfaces [1–7]. We study the harmonic Prym bundle, whose fibers are the spaces of harmonic Prym differentials on variable compact Riemann surfaces and find its connection with Gunning’s cohomological bundle over the Teichmüller space for two important subgroups of the inessential and normalized characters on a compact Riemann surface. We study the periods of holomorphic Prym differentials for essential characters on variable compact Riemann surfaces.     

A Generalization of Fueter’s Theorem

Fueter’s Theorem on the construction of monogenic quaternionic functions starting with a holomorphic function in the upper half of the complex plane, is further generalized in a Clifford analysis setting. The result obtained contains previous generalizations as special cases.     

Rank two quadratic pairs and surface group representations

Let X be a compact Riemann surface. A quadratic pair on X consists of a holomorphic vector bundle with a quadratic form which takes values in a fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the SO₀(2, 3)-character variety of X.     

Division theorems for exact sequences

Under certain integrability and geometric conditions, we prove division theorems for the exact sequences of holomorphic vector bundles and improve the results in the case of Koszul complex. By introducing a singular Hermitian structure on the trivial bundle, our results recover Skoda’s division theorem for holomorphic functions on pseudoconvex domains in complex Euclidean spaces.     

Stolarsky’s invariance principle for projective spaces

We show that Stolarsky’s invariance principle, known for point distributions on the Euclidean spheres, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces

We study magnetic Schrödinger operators on line bundles over Riemann surfaces endowed with metrics of constant curvature. We show that for harmonic magnetic fields the spectral geometry of these operators is completely determined by the Bochner Laplacians of the line bundles. Therefore we are led to examine the spectral problem for the Bochner Laplacian ∇^∗∇ of a Hermitian line bundle L with connection ∇ over a Riemann surface S. This spectral problem is analyzed in terms of the natural holomorphic structure on L defined by the Cauchy-Riemann operator associated with ∇. By means of an elliptic chain of line bundles obtained by twisting L with the powers of the canonical bundle we prove that there exists a certain subset of the spectrum σhol(∇^∗∇) such that the eigensections associated with λ∈σhol(∇^∗∇) are given by the holomorphic sections of a certain line bundle of the elliptic chain. For genus p=0,1 we prove that σhol(∇^∗∇) is the whole spectrum, whereas for genus p>1 we get a finite number of eigenvalues.