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We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions. [ABSTRACT FROM AUTHOR]

A finite group G acts as a group of automorphisms on a compact Riemann surface S of genus g if and only if there exist a Fuchsian group Φ and an epimorphism Φ: Γ→G such that kerΦ = K is a surface group of genus g. And then Φ is named as smooth homomorphism. The objective of this paper is to establish a set of necessary and sufficient conditions for the existence of smooth epimorphism from a Fuchsian group to the finite group of symmetries of Carbon Tetra chloride molecule, whose abstract group representation is a,b|a4= b3=(ab)2?. [ABSTRACT FROM AUTHOR]

Suppose S is a compact oriented surface of genus σ ≥ 2 and C p is a group of orientation preserving automorphisms of S of prime order p ≥ 5 . We show that there is always a finite supergroup G > C p of orientation preserving automorphisms of S except when the genus of S / C p is minimal (or equivalently, when the number of fixed points of C p is maximal). Moreover, we exhibit an infinite sequence of genera within which any given action of C p on S implies C p is contained in some finite supergroup and demonstrate for genera outside of this sequence the existence of at least one C p -action for which C p is not contained in any such finite supergroup (for sufficiently large σ ). [ABSTRACT FROM AUTHOR]

In this paper we extend vessel theory, or equivalently, the theory of overdetermined 2 D systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beurling type theorem on indefinite Hardy spaces on finite bordered Riemann surfaces. [ABSTRACT FROM AUTHOR]

Saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector-valued critical measures , a notion that finds a natural motivation in several branches of analysis. We study in depth the case of measures μ → = ( μ 1 , μ 2 , μ 3 ) when the mutual interaction comprises both attracting and repelling forces. For arbitrary vector polynomial external fields we establish general structural results about critical measures, such as their characterization in terms of an algebraic equation solved by an appropriate combination of their Cauchy transforms, and the symmetry properties (or the S -properties) exhibited by such measures. In consequence, we conclude that vector-valued critical measures are supported on a finite number of analytic arcs, that are trajectories of a quadratic differential globally defined on a three-sheeted Riemann surface. The complete description of the so-called critical graph for such a differential is the key to the construction of the critical measures. We illustrate these connections studying in depth a one-parameter family of critical measures under the action of a cubic external field. This choice is motivated by the asymptotic analysis of a family of (non-hermitian) multiple orthogonal polynomials, that is subject of a forthcoming paper. Here we compute explicitly the Riemann surface and the corresponding quadratic differential, and analyze the dynamics of its critical graph as a function of the parameter, giving a detailed description of the occurring phase transitions. When projected back to the complex plane, this construction gives us the complete family of vector-valued critical measures, that in this context turn out to be vector-valued equilibrium measures. [ABSTRACT FROM AUTHOR]

Let ( X, q) be an elliptic curve marked at the origin. Starting from any cover π: Γ → X of an elliptic curve X marked at d points { π } of the fiber π ( q) and satisfying a particular criterion, Krichever constructed a family of d × d matrix KP solitons, that is, matrix solutions, doubly periodic in x, of the KP equation. Moreover, if Γ has a meromorphic function f: Γ → P with a double pole at each p , then these solutions are doubly periodic solutions of the matrix KdV equation U = 1/4(3 UU + 3 U U + U ). In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the d points { p }; i.e. Γ is hyperelliptic and each pi is a Weierstrass point of Γ. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker-Akhiezer function of X; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons. [ABSTRACT FROM AUTHOR]

We investigate the behavior of the real part of the logarithmic derivatives of the Selberg zeta-functionsZPSL(2,Z)(s) andZC(s) in the critical strip 0< σ <1. The functionsZPSL(2,Z)(s) andZC(s) are defined on the modular group and on the compact Riemann surface, respectively. [ABSTRACT FROM AUTHOR]

It is proved that Galois coverings of smooth families of compact Riemann surfaces over Stein manifolds are holomorphically convex if the covering transformation groups are isomorphic to discrete subgroups of the automorphism group of the unit disc. The proof is based on an extension of the fact that disc bundles over compact Kähler manifolds are weakly 1-complete. [ABSTRACT FROM AUTHOR]

The theory of multiplicative functions and Prym differentials on a compact Riemann surface has found numerous applications in function theory, analytic number theory, and equations of mathematical physics. We give a full constructive description for the divisors of elementary abelian differentials of integer order and all three kinds depending holomorphically on the modulus of compact Riemann surfaces F. We study the location of zeros of holomorphic Prym differentials on F, as well as the structure of the set of (multiplicatively) special divisors on F in the spaces F and F. [ABSTRACT FROM AUTHOR]

A Calabi's theorem says that on a compact Riemann surface, an extremal metric is a constant scalar curvature metric. In this paper, we use a new method to prove this theorem. Then we give an interesting corollary. [ABSTRACT FROM AUTHOR]

We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic C. This can be expressed as a relation between the period spectrum and the ortholength spectrum of C. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along C as well estimates on the lengths of geodesic segments which start and end orthogonally on C. Variant trace formulas also lead to several simultaneous nonvanishing results for different periods. [ABSTRACT FROM AUTHOR]

Abstract: We present some results on two meromorphic functions from S to sharing a number of values where S is a Riemann surface of one of the following types: compact, compact minus finitely many points, the unit disk, a torus, the complex plane. [Copyright &y& Elsevier]

We study the self-dual Chern–Simons Higgs equation on a compact Riemann surface with Neumann boundary condition. We show that the Chern–Simons Higgs equation with parameter λ > 0 has at least two solutions $(u_{\lambda}^{1}, u_{\lambda}^{2})$ for λ sufficiently large, such that $u_{\lambda}^{1}\rightarrow-u_0$ almost everywhere as λ → + ∞, and that $u_{\lambda}^{2}\rightarrow -\infty$ almost everywhere as λ → ∞, where u0 is a (negative) Green function on M. [ABSTRACT FROM AUTHOR]

Abstract: If p is prime, a compact Riemann surface X of genus is called cyclic p-gonal if it admits a cyclic group of automorphisms of order p such that the quotient space has genus 0. If in addition is not normal in the full automorphism A, then we call X a non-normal p-gonal surface. In the following we classify all non-normal p-gonal surfaces. [Copyright &y& Elsevier]

Let X be a compact Riemann surface of genus g>1 and let G be a group of biholomorphic mappings on X onto itself. Consider all pairs (X,G). We say that (X,G) is topologically equivalent to (X′,G′) if there exists an o.p. (orientation preserving) homeomorphism h of X onto X′ such that G′h=hG. In this paper, we shall classify the (X,G)''s up to topological equivalence in the case g=4. [Copyright &y& Elsevier]

Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are sections of a line bundle of multiplicative half-order differentials on a compact real Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface. [ABSTRACT FROM AUTHOR]

Let X be a smooth and connected Riemann surface of genus g ≥ 0 and ƒ : X ↵ P¹, h : X ↵ P¹ non-constant meromorphic functions on X. Fix an integer n ≥ 4 and assume the existence of n distinct points a1,...,an ϵ P¹ such that ƒ-1(ai) = h-1(ai) (set-theoretically) for every i. Here we prove that either ƒ=h or (n - 4)(deg(ƒ) + deg(h)) ≤2g - 2. [ABSTRACT FROM AUTHOR]