Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.     

The Geometry of Two Generator Groups: Hyperelliptic Handlebodies

A Kleinian group naturally stabilizes certain subdomains and closed subsets of the closure of hyperbolic three space and yields a number of different quotient surfaces and manifolds. Some of these quotients have conformal structures and others hyperbolic structures. For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a hyperelliptic three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the Weierstrass lines of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six generalized Weierstrass points on the surface. In addition, on each of these equidistant surfaces we find an orientation reversing involution that fixes curves through the generalized Weierstrass points.Mathematics Subject Classifications (2000). primary 30F10, 30F35, 30F40; secondary 14H30, 22E40.     

On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus

We obtain an upper bound for the number of holomorphic mappings of a genus 3 Riemann surface onto a genus 2 Riemann surface in a series of cases. In particular, we establish that the number of holomorphic mappings of an arbitrary genus 3 Riemann surface onto an arbitrary genus 2 Riemann surface is at most 48. We show that this estimate is sharp and find pairs of Riemann surfaces for which it is attained.     

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.     

Minimal genus problem for pseudo-real Riemann surfaces

A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.     

Uniqueness of the Riemann minimal examples

We prove that a properly embedded minimal surface in R ³ of genus zero with infinite symmetry group is a plane, a catenoid, a helicoid or a Riemann minimal example. We introduce the language of Hurwitz schemes to understand the underlying moduli space of surfaces in our setting. Oblatum 30-V-1997 & 5-VIII-1997     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

On Real Forms of Belyi Surfaces With Symmetric Groups of Automorphisms

In virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study the actions of the symmetric groups S _n on Belyi Riemann surfaces. We show that such surfaces are symmetric and we calculate the number of connected components of the corresponding real forms.     

Isomonodromy Approach to Boundary Value Problems for the Ernst Equation

We use the isomonodromy properties of theta-functional solutions of the Ernst equation and an asymptotic expansion in the spectral parameter to establish algebraic relations, enforced by the underlying Riemann surface, between the metric functions and their derivatives. These relations determine which classes of boundary value problems can be solved on a given surface. The situation on lower-genus Riemann surfaces is studied in detail.     

Boundary problems of the theory of analytic functions on Riemann surfaces

The survey includes papers reviewed in RZh Matematika from 1954–1979. We consider the Riemann boundary problem on a compact Riemann surface and in the class of piecewise-meromorphic automorphic functions; singular integral equations with automorphic kernels and in the form of Abelian integrals; the method of symmetry in solving the problems of Hilbert (linear and nonlinear), Schwarz, Carleman, etc., in the case of a Riemann surface with boundary and in the case of a planar domain, bounded by an algebraic curve; and boundary problems on open Riemann surfaces.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 18, pp. 3–66, 1980.     

A foliated analogue of one- and two-dimensional Arakelov theory

The analogy between number fields and Riemann surfaces was an important source of motivation for mathematicians in the last century. We improve and extend this analogy by substituting Riemann surfaces with certain foliations by Riemann surfaces. In particular we show that coverings of these foliations lead to formulas having the same structure as formulas describing number field extensions. We also study higher dimensional foliations which have properties analogous to arithmetic surfaces. This provides more evidence for a conjecture of Deninger.     

Inverse problem of hydrodynamics for doubly connected domain

We consider an inverse problem of hydrodynamics for flow past pair of aerofoils. We find a general form of its solution. The key part of problem’s solving is to determine numerical parameters defining flow domain and complex velocity in it up to conformal mapping (the parameters problem). The solvability of parameters problem is proved for various flow schemes. For that we essentially use the interpretation of the problem in terms of Riemann surface and the Riemann surfaces theory.     

Once-holed tori embedded in Riemann surfaces

Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and consitute a partially ordered set, the order being defined in terms of conforaml embeddings. We consider some families of once-holed tori that are conformally embedded in target Riemann surfaces of conformal mappings of a given noncompact Riemann surface of genus one, and establish an analogue of the one-quarter theorem of Koebe. We also investigate families of once-holed tori conformally embedded in a Riemann surface of positive genus.      

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.     

Corrections and complements to ��Once-holed tori embedded in Riemann surfaces��

Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and can be used to measure the sizes of handles of Riemann surfaces of positive genus. We study some families of once-holed tori that are conformally embedded in target Riemann surfaces of conformal mappings of a given noncompact Riemann surface of genus one, and correct some results given in Masumoto (Math. Z. 257:453?C464, 2007).     

Transplantation and Jacobians of Sunada isospectral Riemann surfaces

We consider relations among the Jacobians of isospectral compact Riemann surfaces constructed using Sunada's theorem. We use a simple algebraic formulation of “transplantation” of holomorphic 1-forms and singular 1-cycles to obtain two main results. First, we obtain a geometric proof of a result of Prasad and Rajan that Sunada isospectral Riemann surfaces have isogenous Jacobians. Second, we determine a relationship (weaker than isogeny) that holds among the Jacobians of Sunada isospectral Riemann surfaces when the Jacobians’ extra structure as principally polarized abelian varieties is taken into account. We also show all Sunada isospectral manifolds have isomorphic real cohomology algebras. Finally, we exhibit transplantation of cycles explicitly in a concrete example of a pair of isospectral Riemann surfaces constructed by Brooks and Tse.     

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.     

Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals

The known examples of explicit equations for Riemann surfaces whose field of moduli is different from their field of definition, are all hyperelliptic. In this paper we construct a family of equations for non-hyperelliptic Riemann surfaces, each of them is isomorphic to its conjugate Riemann surface, but none of them admit an anticonformal automorphism of order 2; that is, each of them has its field of moduli, but not a field of definition, contained in \mathbb R{{\mathbb R}} . These appear to be the first explicit such examples in the non-hyperelliptic case.     

Bergman interpolation on finite Riemann surfaces. Part I: asymptotically flat case

The article considers the Bergman space interpolation problem on open Riemann surfaces obtained from a compact Riemann surface by removing a finite number of points. Such a surface is equipped with what we call an asymptotically flat conformal metric, i.e., a complete metric with zero curvature outside a compact subset. Sufficient conditions for interpolation in weighted Bergman spaces over asymptotically flat Riemann surfaces are then established. When the weights have curvature that is quasi-isometric to the asymptotically flat boundary metric, these sufficient conditions are shown to be necessary, unless the surface has at least one cylindrical end, in which case, the necessary conditions are slightly weaker than the sufficient conditions.     

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.