On cyclic <Emphasis Type="Italic">p</Emphasis>-gonal Riemann surfaces with several <Emphasis Type="Italic">p</Emphasis>-gonal morphisms

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.     

Normal families of BMOloc-quasiregular mappings

Given a function Q(z) of locally bounded mean oscillation in a Riemann surface X, we prove a normality criterion for a family of Q(z)-quasiregular mappings between two homeomorphic Riemann surfaces X, Y, normalized by the condition that the preimages of two given points be two fixed points. Several examples and counter-examples are included.     

A sewing theorem for quadratic differentials

Quadratic differentials \mathfrakQ(z)dz² \mathfrak{Q}(z)d{z^2} on a finite Riemann surface with poles of order not exceeding two are considered. The existence of such a differential with prescribed metric characteristics is proved. These characteristics are the following: the leading coefficients in the expansions of the function \mathfrakQ(z) \mathfrak{Q}(z) in neighborhoods of its poles of order two, the conformal modules of the ring domains, and the heights of the strip domains in the decomposition of the Riemann surface defined by the structure of trajectories of this differential. Bibliography: 5 titles.     

Total curvature and volume of foliations on the sphere S 2

In this paper we study a curvature integral associated with a pair of orthogonal foliations on the Riemann sphere S ² and we compute the minimal value of the volume of meromorphic foliations.     

A complete conformal metric of preassigned negative Gaussian curvature for a punctured hyperbolic Riemann surface

Leth be a complete metric of Gaussian curvature K₀ on a punctured Riemann surface of genusg ≥ 1 (or the sphere with at least three punctures). Given a smooth negative functionK withK =K ₀ in neighbourhoods of the punctures we prove that there exists a metric conformal toh which attains this function as its Gaussian curvature for the punctured Riemann surface. We do so by minimizing an appropriate functional using elementary analysis.     

Triangulations and moduli spaces of Riemann surfaces with group actions

Summary We study that subset of the moduli space of stable genusg,g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite groupF acts. We show first that this subset is compact. It turns out that, for general finite groupsF, the above subset is not connected. We show, however, that for ℤ₂ actions this subsetis connected. Finally, we show that even in the moduli space ofsmooth genusg Riemann surfaces, the subset of those Riemann surfaces on which ℤ₂ actsis connected. In view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certainregular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically. This work has been supported by the European Communities Science Plan Project No SCI^*-CT91 (TSTS) “Computational Methods in the Theory of Riemann Surfaces and Algebraic Curves,” by Academy of Finland and by the Swiss National Science Foundation Grant 20-34099.92. We thank M. C. Petrus for providing excellent motivation for this work. This article was processed by the author using the LATEX style filecljourl from Springer-Verlag.     

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Using coherent-state techniques, we prove a sampling theorem for Majorana’s (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of N samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to J, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.      

On lengths of filling closed geodesics of hyperbolic punctured Riemann spheres

Let S be a Riemann sphere with n ≥ 4 points deleted. In this article we investigate certain filling closed geodesics of S and give quantitative common lower bounds for the hyperbolic lengths of those geodesics with respect to any hyperbolic structure on S (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spline approximation of a non-linear Riemann–Hilbert problem†

In this article we use linear spline approximation of a non-linear Riemann–Hilbert problem on the unit disk. The boundary condition for the holomorphic function is reformulated as a non-linear singular integral equation A(u) = 0, where A : H ¹(Γ) → H ¹(Γ) is defined via a Nemytski operator. We approximate A by A ⁿ : H ¹(Γ) → H ¹(Γ) using spline collocation and show that this defines a Fredholm quasi-ruled mapping. Following the results of (A.I. ?nirel'man, The degree of quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sbornik 18 (1972), pp. 373–396; M.A. Efendiev, On a property of the conjugate integral and a nonlinear Hilbert problem, Soviet Math. Dokl. 35 (1987), pp. 535–539; M.A. Efendiev, W.L. Wendland, Nonlinear Riemann–Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), pp. 37–58; Nonlinear Riemann–Hilbert problems without transversality. Math. Nachr. 183 (1997), pp. 73–89; Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), pp. 111–124; Nonlinear Riemann–Hilbert problems with Lipschitz, continuous boundary data without transversality, Nonlinear Anal. 47 (2001), pp. 457–466; Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data: Doubly connected domains, Proc. Roy. Soc. London Ser. A 459 (2003), pp. 945–955.), we define a degree of mapping and show the existence of the spline solutions of the fully discrete equations A ⁿ (u) = 0, for n large enough. We conclude this article by discussing the solvability of the non-linear collocation method, where we shall need an additional uniform strong ellipticity condition for employing the spline approximation.     

Quotients of Incidence Algebras and the Euler Characteristic

In this note, we show that, if A ? kQ _A /I _A is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with Q _A , then A ? kΣ if and only if I _A has a generating set consisting of exactly χ(Q _A ) elements, where χ(Q _A ) is the Euler characteristic of Q _A . We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected.     

Field of definition and Galois orbits for the Macbeath-Hurwitz curves

Hurwitz curves are Riemann surfaces with 84(g-1) automorphisms, g the genus. Defined over some number field they permit an obvious Gal ([`(\Bbb Q)]/\Bbb Q){\rm Gal} (\overline {{\Bbb Q}}/{\Bbb Q}) action. We investigate this action for the first known infinite series of Hurwitz curves, due to Macbeath, using the canonical model of the curves. As a result we obtain the minimal field of definition for these curves. The method can be extended to some other infinite series of modular curves for non-congruence subgroups.     

On Involutive Cluster Automorphisms

We construct a special embedding of the translation quiver ?Q in the three-dimensional affine space 𝔸³ where Q is a finite connected acyclic quiver and ?Q contains a local slice whose quiver is isomorphic to the opposite quiver Q ^op of Q. Via this embedding, we show that there exists an involutive anti-automorphism of the translation quiver ?Q. As an immediate consequence, we characterize explicitly the group of cluster automorphisms of the cluster algebras of seed (X, Q), where Q and Q ^op are mutation equivalent.     

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

Estimates of sums of zero multiplicities for eigenfunctions of the Laplace-Beltrami operator

We obtain an upper estimate N?_χ(M) for the sum Q _N of singular zero multiplicities of the Nth eigenfunction of the Laplace-Beltrami operator on the two-dimensional, compact, connected Riemann manifold M, where _χ M is the Euler characteristic ofM. Stronger estimates, but equivalent asymptotically (N å ∞), are given for the cases of the sphere S ² and the projective plane ?². Asymptotically sharper estimates are shown for the case of a domain on the plane.     

On the Null Cone for Quiver Representations Under Reflection Functors

Let d be a prehomogeneous dimension vector for a connected quiver Q with the property that c ^? d has a negative entry for some ? ∈ ? where c is the Coxeter transformation corresponding to an admissible numbering of the vertices Q ₀ of Q. Denote by rep(Q, d) the variety of d-dimensional representations of Q and by Sl(d) the product of the special linear groups at all vertices Q ₀. We show how to find the irreducible components of the null cone of the algebraic quotient rep(Q, d)//Sl(d).     

On a class of conformal metrics arising in the work of Seiberg and Witten

We examine a class of conformal metrics arising in the “N = 2 supersymmetric Yang-Mills theory” of Seiberg and Witten. We provide several alternate characterizations of this class of metrics and proceed to examine issues of existence and boundary behavior and to parameterize the collection of Seiberg-Witten metrics with isolated non-essential singularities on a fixed compact Riemann surface. In consequence of these results, the Riemann sphere does not admit a Seiberg-Witten metric, but for all there is a conformal metric on of regularity which is Seiberg-Witten off of a finite set. Received August 18, 1998     

de Sitter空间S1m+1中具有平行Blaschke张量的正则类空超曲面

本文引入两个以de Sitter空间为模型的非齐性坐标来覆盖共形空间Q₁^m+1.利用球面S^m+1中超曲面的Möbius几何的方法,本文研究了Q₁^m+1中正则类空超曲面的共形几何.作为其结果,本文对所有具有平行Blaschke张量的正则类空超曲面进行了完全分类.     

Harmonic maps into loop spaces of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. We conjecture that harmonic maps of the Riemann sphere ℂℙ¹ into ΩG are related to Yang-Mills G-fields on ℝ⁴. This work was partly supported by the RFBR (Grant Nos. 04-01-00236, 06-02-04012), by the program of Support of Scientific Schools (Grant No. 1542.2003.1), and by the Scientific Program of RAS “Nonlinear Dynamics”     

Matrix Riemann-Hilbert problems and factorization on Riemann surfaces

The Wiener-Hopf factorization of 2×2 matrix functions and its close relation to scalar Riemann-Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q₁,Q₂) is defined. To each class C(Q₁,Q₂) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q₁,Q₂) is reduced to solving a scalar Riemann-Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems.