The spaces of meromorphic Prym differentials on a finite Riemann surface

In the previous articles the second author started constructing a general theory of multiplicative functions and Prym differentials on a compact Riemann surface for arbitrary characters. Function theory on compact Riemann surfaces differs substantially from that on finite Riemann surfaces. In this article we start constructing a general function theory on variable finite Riemann surfaces for multiplicative meromorphic functions and differentials. We construct the forms of all elementary Prym differentials for arbitrary characters and find the dimensions of, and also construct explicit bases for, two important quotient spaces of Prym differentials. This yields the dimension of and a basis for the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters.     

Periods of Harmonic Prym Differentials on a Compact Riemann Surface

We study the period classes of closed, harmonic, and holomorphic Prym differentials on a compact Riemann surface of any genus g2 for arbitrary characters of its fundamental group. We prove that the harmonic Prym vector bundle of harmonic Prym differentials and the Gunning cohomology bundle are real-analytically isomorphic over the base of nontrivial normalized characters for every compact Riemann surface of genus g2.     

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.     

Prym differentials on a variable compact Riemann surface

The dimensions and bases of the spaces of meromorphic Prym differentials on a variable compact Riemann surface, as well as in the first holomorphic de Rham cohomology group of Prym differentials, are found for any characters.     

Prym differentials as solutions to boundary value problems on Riemann surfaces

Construction of multiplicative functions and Prym differentials, including the case of characters with branch points, reduces to solving a homogeneous boundary value problem on the Riemann surface. The use of the well-established theory of boundary value problems creates additional possibilities for studying Prym differentials and related bundles. Basing on the theory of boundary value problems, we fully describe the class of divisors of Prym differentials and obtain new integral expressions for Prym differentials, which enable us to study them directly and, in particular, to study their dependence on the point of the Teichmüller space and characters. Relying on this, we obtain and generalize certain available results on Prym differentials by a new method.     

Weierstrass Multiplicative Points on a Compact Riemann Surface

We introduce Weierstrass multiplicative points and develop the theory of Weierstrass multiplicative points for multiplicative meromorphic functions and Prym differentials on a compact Riemann surface. We prove some analogs of the Weierstrass and Noether theorems on the gaps of multiplicative functions. We obtain two-sided estimates for the number of Weierstrass multiplicative points and q-points. We propose a method for studying the Weierstrass and Noether gaps and Weierstrass multiplicative points by means of filtrations in the Jacobi variety of a compact Riemann surface.     

An analogue of the rauch variational formula for prym differentials

In [5], H. E. Rauch discovered a formula for the first variation of an abelian differential on a Riemann surface and its periods with respect to the change of complex structure induced by a Beltrami differential. R. S. Hamilton, in [3], and discussed by C. Earle in [1], found an elegant proof of the formula using only first principles and not requiring uniformization theory. His proof uses a small amount of Hodge theory, the Riemann bilinear period relations, and a simple operator construction. In this article, we find an analogue of Rauch’s formula for the Prym differentials using some of Hamilton’s techniques, the Hodge theorem for vector bundles, and the “Prym version” of the Riemann bilinear relations. We discover a complicated set of formulas for the variation of the Prym differentials, with different specific solutions depending to the make-up of the Prym character. We conclude that the variation of the Prym periods with a given character depends on the differentials for the character and the differentials for its inverse. This explains the simplicity of the classical case, where the character is its inverse.     

Meromorphic differentials with twisted coefficients on compact Riemann surfaces

We apply the Dirichlet’s principle to a modified energy functional on Riemann surfaces to reprove the existence of harmonic metrics with certain prescribed singularities due to Simpson, Sabbah and Biquard–Boalch, and hence of differentials with twisted coefficients of the second and third kinds. As a by-product, this generalizes the classical theory of Abelian differentials on a compact Riemann surface to the case of twisted coefficients. This also proposes a more natural approach for general existence of harmonic metrics in the higher dimensional case. The author supported partially by NSF of China (No. 10471105, 10771160).     

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.     

Multiplicative Weierstrass Points and the Jacobi Variety of a Compact Riemann Surface

In the previous papers of the present author, a general theory of multiplicative Weierstrass points on compact Riemann surfaces for arbitrary characters was developed. In the present paper, some additional relations between multiplicative Weierstrass points on a compact Riemann surface for an arbitrary character and special subsets in the Jacobi variety, the canonical embedding of a compact Riemann surface into a projective space, are established. We not distinction between classical Weierstrass points and multiplicative Weierstrass points on a compact Riemann surface.     

Compactification of moduli space of harmonic mappings

We introduce the notion of harmonic nodal maps from the stratified Riemann surfaces into any compact Riemannian manifolds and prove that the space of the energy minimizing nodal maps is sequentially compact. We also give an existence result for the energy minimizing nodal maps. As an application, we obtain a general existence theorem for minimal surfaces with arbitrary genus in any compact Riemannian manifolds. Received: 1 April 1997; revised: 15 April 1998.     

Asymptotics of the Teichmüller harmonic map flow

The Teichmüller harmonic map flow, introduced by Rupflin and Topping (2012)  [11], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.     

On the Teichmüller theorem and the heights theorem for quadratic differentials

By using the Marden-Strebel heights theorem for quadratic differentials, we provide a concrete method for finding the Teichmüller differential associated with the Teichmüller mapping between compact or finitely punctured Riemann surfaces.

     

Stability of rank 2 vector bundles along isomonodromic deformations

We are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact Riemann surfaces, which are provided with irreducible meromophic tracefree connections. In the case of a logarithmic connection on the Riemann sphere, such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles, according to a result of A. Bolibruch. In the general case of meromorphic connections over Riemann surfaces of arbitrary genus, we prove that the vector bundle will be semi-stable, up to a small isomonodromic deformation. More precisely, the vector bundle underlying the universal isomonodromic deformation is generically semi-stable along the deformation, and even maximally stable. For curves of genus g ≥ 2, this result is non-trivial even in the case of non-singular connections. The author was partially supported by ANR SYMPLEXE BLAN06-3-137237.     

Asymptotic Windings of Brownian Motion Paths on Riemann Surfaces

We study asymptotic winding properties of Brownian motion paths on Riemann surfaces by obtaining limit laws for stochastic line integrals along Brownian paths of meromorphic differential 1-forms (Abelian differentials).     

The zero-free region of the derivative of Selberg zeta functions

In this article, we study the zero-free region of the derivative of Selberg zeta functions associated with compact Riemann surfaces and three-dimensional compact hyperbolic spaces. We obtain the zero-free region with respect to the left of the critical line of each Selberg zeta function. This is an improvement of Wenzhi Luo’s zero-free region theorem for compact Riemann surfaces.     

The zero-free region of the derivative of Selberg zeta functions

In this article, we study the zero-free region of the derivative of Selberg zeta functions associated with compact Riemann surfaces and three-dimensional compact hyperbolic spaces. We obtain the zero-free region with respect to the left of the critical line of each Selberg zeta function. This is an improvement of Wenzhi Luo’s zero-free region theorem for compact Riemann surfaces.     

Real meromorphic differentials: a language for describing meron configurations in planar magnetic nanoelements

We use the language of real meromorphic differentials from the theory of Klein surfaces to describe the metastable states of multiply connected planar ferromagnetic nanoelements that minimize the exchange energy and have no side magnetic charges. These solutions still have sufficient internal degrees of freedom, which can be used as Ritz parameters to minimize other contributions to the total energy or as slow dynamical variables in the adiabatic approximation. The nontrivial topology of the magnet itself leads to several effects first described for the annulus and observed in the experiment. We explain the connection between the numbers of topological singularities of various types in the magnet and the constraints on the positions of these singularities following from the Abel theorem. Using multivalued Prym differentials leads to new meron configurations that were not considered in the classic work by Gross.     

A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials

In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of uniform thickness of hyperbolic Riemann surfaces in terms of invariant metrics.