Finite Riemann Surfaces Research Articles

Selberg Zeta function and hyperbolic Eisenstein series

Let [Formula: see text] be a Fuchsian group acting on the hyperbolic upper half-plane [Formula: see text], such that [Formula: see text] is a geometrically finite Riemann surface with respect to the natural hyperbolic metric induced from [Formula: see text]. If [Formula: see text] is hyperbolic then following [J. Kudla and S. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979) 193–211; J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293 (1977) 143–203], there is a corresponding hyperbolic Eisenstein series. In this paper, we study the limiting behavior of hyperbolic Eisenstein series on a degenerating family of geometrically finite hyperbolic surfaces. In particular, we give a partial lightening to a question of Ji, concerning the approximation of Eisenstein series during degeneration (see Proposition 5.2 and Theorem 5.2).

The quasiconformal equivalence of Riemann surfaces and the universal Schottky space

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite-type, the situation is rather complicated. In this paper, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to obtain a universal property of the deformation spaces of Schottky regions, which is analogous to the fact that the universal Teichmüller space contains all Teichmüller spaces.

A classification problem on mapping classes on fiber spaces over Teichmüller spaces

Let $\tilde{S}$ be an analytically finite Riemann surface which is equipped with a hyperbolic metric. Let $S=\tilde{S}\backslash \{\mbox{one point} x\}$. There exists a natural projection $\Pi$ of the $x$-pointed mapping class group Mod$_S^x$ onto the mapping class group Mod$(\tilde{S})$. In this paper, we classify elements in the fiber $\Pi^{-1}(\chi)$ for an elliptic element $\chi\in \mbox{Mod}(\tilde{S})$, and give a geometric interpretation for each element in $\Pi^{-1}(\chi)$. We also prove that $\Pi^{-1}(t_a^n\circ \chi)$ or $\Pi^{-1}(t_a^n\circ \chi^{-1})$ consists of hyperbolic mapping classes provided that $t_a^n\circ \chi$ and $t_a^n\circ \chi^{-1}$ are hyperbolic, where $a$ is a simple closed geodesic on $\tilde{S}$ and $t_a$ is the positive Dehn twist along $a$.

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.

ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

THE RESIDUE THEOREM AND AN ANALOG OF P. APPELL’S FORMULA FOR FINITE RIEMANN SURFACES

A theory of multiplicative functions and Prym differentials for the case of special characters on compact Riemann surfaces has found applications in geometrical function theory of complex variable, analytical number theory and in equations of mathematical physics. Theory of functions on compact Riemann surfaces differs from the theory of functions on finite Riemann surfaces even for the class of single meromorphic functions and Abelian differentials. In this article we continue the construction of the general function theory on finite Riemann surfaces for multiplicative meromorphic functions and differentials. We have proved analogues of the theorem on the full sum of residues for Prym differentials of every integral order and P. Appell's formula on expansion of the multiplicative function with poles of arbitrary multiplicity in the sum of elementary Prym integrals.

The real analog of the Jacobi inversion problem on a Riemann surface with boundary, its generalizations, and applications

Given a finite Riemann surface of genus h ≥ 1 with boundary composed of m+1 connected components we consider a system of m+h real congruences analogous to the classical Jacobi inversion problem. We provide a solution to this system and its applications to boundary value problems.

Single-Valued q-Differentials on a Variable Finite Riemann Surface

We consider single-valued meromorphic functions and differentials on finite Riemann surfaces. We construct elementary q-differentials of all types. We find the dimension of two important quotient spaces of Abelian q-differentials, in particular, the first holomorphic de Rham cohomology group of 1-differentials. For these quotient spaces we construct bases that locally holomorphically depend on the moduli of the Riemann surfaces. Bibliography: 6 titles.

Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

We formulate the Bergman-type interpolation problem on finite open Riemann surfaces covered by the unit disk. Our version of the interpolation problem generalizes Bergman-type interpolation problems previously studied by Seip, Berndtsson, Ortega Cerda, and a number of other authors. We then prove sufficient conditions for a sequence to be interpolating. When the curvature of the weight in question is bounded in an appropriate sense, we show that the sufficient conditions are almost necessary, but not quite. The results extend work of Ortega Cerda, who resolved the case in which the boundary of the surface is pure 1-dimensional. Our version of the interpolation problem effectively changes the geometry of the underlying space near the 0-dimensional boundary components, or punctures, thereby linking in a crucial way with the previous article (Varolin, J d’Anal Math, 2015, to appear) in this two-part series.

Prym differentials with matrix characters on a finite Riemann surface

The theory of multiplicative functions and Prym differentials for scalar characters on a compact Riemann surface has found applications in function theory, analytic number theory, and mathematical physics.

SONLU RIEMANN YÜZEYLERİNİN KONFORMAL DÖNÜŞÜMLERİ

The purpose of this work is to obtain a new type of conformal mappings of compact finite Riemann surfaces bounded by finitely many analytic Jordan curves. This is achieved by making use of Riemann-Roch theorem. As is well-known, every plane region is conformally equivalent to a parallel slit region. This theorem was carried over the case of Riemann surfaces with finite genus. The other types of conformal mappings can be found in the different literatures. It will be now deal with a different conformal mapping from those. It is a finite sheeted covering surface of the extended complex plane whose each boundary component consists of a closed interval on real axis.

On the length spectrum metric in infinite dimensional Teichmüller spaces

We consider the length spectrum metric dL in infinite dimensional Teichmuller space T(R0). It is known that dL defines the same topology as that of the Teichmuller metric dT on T(R0) if R0 is a topologically finite Riemann surface. In 2003, Shiga proved that dL and dT define the same topology on T(R0) if R0 is a topologically infinite Riemann surface which can be decomposed into pairs of pants such that the lengths of all their boundary components except punctures are uniformly bounded by some positive constants from above and below. In this paper, we extend Shiga's result to Teichmuller spaces of Riemann surfaces satisfying a certain geometric condition.

Patterson-Sullivan-type measures on Riccati foliations with quasi-Fuchsian holonomy

We consider Riccati foliations ℱρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π1(S) → PSL(2,ℂ) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ+ and µ− for positive and negative times, respectively. These measures have a common projection to a harmonic measure μρ for the Riccati foliation. We describe μρ+, μρ- and μρ in terms of the Patterson-Sullivan construction, and we show that the measures μρ provide examples of the conformal harmonic measures introduced by M. Brunella.

On the Isomorphism Question for Complete Pick Multiplier Algebras

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra $${\mathcal{M}_V = \{f \big|_V : f \in \mathcal{M}_d\}}$$ , where d is some integer or $${\infty, \mathcal{M}_d}$$ is the multiplier algebra of the Drury-Arveson space $${H^2_d}$$ , and V is a subvariety of the unit ball. For finite dimensional d it is known that, under mild assumptions, every isomorphism between two such algebras $${\mathcal{M}_V}$$ and $${\mathcal{M}_W}$$ is induced by a biholomorphism between W and V. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where V is the proper image of a finite Riemann surface. The second deals with the case where V is a disjoint union of varieties.

Extending the Schwarz-Christoffel Formula to Universal Covering Maps of a Riemann Surface

The classical Schwarz-Christoffel formula allows one to compute the conformal map of the unit disc onto a domain bounded by a polygon. An extension of this formula can be obtained that allows one to compute the conformal map of the unit disc onto a domain bounded by a “quadratic differential polygon” i.e. a closed Jordan curve that is the union of trajectory arcs of a certain quadratic differential and their end points. Now suppose that R is a finite Riemann surface with boundary and π: D → R is a universal covering map of R. Let Q(z) dz2 be a quadratic differential on R and let R be a sub-Riemann surface of R such that the boundary of R is the union of finitesided Q(z) dz2 polygons. If the lift of R, D = π- 1(R), is a simply-connected domain, then the Riemann mapping theorem ensures that there is a conformal map f of the unit disc onto D. In this paper, we will prove a generalization of the Schwarz-Christoffel formula that allows us to compute f. Then π ◦f is a universal covering map of R.

Teichmüller disks and hyperbolic mapping classes

Let $\tilde{S}$ be an analytically finite Riemann surface of type $(p,n)$ with $3p+n>3$. Let $x\in \tilde{S}$ and $S=\tilde{S}\backslash \{x\}$. Let $\mbox{Mod}_S^x$ denote the $x$-pointed mapping class group of $S$ and $\mbox{Mod}_{\tilde{S}}$ the mapping class group of $\tilde{S}$. Then the natural projection $J:T(S)\rightarrow T(\tilde{S})$ between Teichmüller spaces induces a group epimorphism $I:\mbox{Mod}_S^x\rightarrow \mbox{Mod}_{\tilde{S}}$. It is well known that for a given Teichmüller disk $\tilde{\Delta}$ in $T(\tilde{S})$, there is a family $\mathscr{F}(\tilde{\Delta})$ of Teichmüller disks $\Delta(z)$ in $T(S)$ parametrized by a hyperbolic plane. If $\tilde{\Delta}$ is invariant under a hyperbolic mapping class $\tilde{\theta}$, then all known hyperbolic mapping classes $\theta\in \mbox{Mod}_S^x$ for which $I(\theta)=\tilde{\theta}$ stem from the construction of $\mathscr{F}(\tilde{\Delta})$. We show that if $\tilde{\theta}$ is represented by a product of Dehn twists along two filling simple closed geodesics, then there exist infinitely many hyperbolic mapping classes $\gamma\in \mbox{Mod}_S^x$ with $I(\gamma)=\tilde{\theta}$ so that their invariant Teichmüller disks are not members of $\mathscr{F}(\tilde{\Delta})$. The result contrasts with the original pattern established by I. Kra.

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.

The fiber product of Riemann surfaces: a Kleinian group point of view

Let P 1 : S 1 → S and P 2 : S 2 → S be non-constant holomorpic maps between closed Riemann surfaces. Associated to the previous data is the fiber product \({S_{1} \times_{S} S_{2}=\{(x,y) \in S_{1} \times S_{2}: P_{1}(x)=P_{2}(y)\}}\) . This is a compact space which, in general, fails to be a Riemann surface at some (possible empty) finite set of points \({F \subset S_{1} \times_{S} S_{2}}\) . One has that S 1 × S S 2 − F is a finite collection of analytically finite Riemann surfaces. By filling out all the punctures of these analytically finite Riemann surfaces, we obtain a finite collection of closed Riemann surfaces; whose disjoint union is the normal fiber product \({\widetilde{S_{1}\times_{S} S_{2}}}\) . In this paper we prove that the connected components of \({\widetilde{S_{1}\times_{S} S_{2}}}\) of lowest genus are conformally equivalents if they have genus different from one (isogenous if the genus is one). A description of these lowest genus components are provided in terms of certain class of Kleinian groups; B-groups.

A sewing theorem for quadratic differentials

Quadratic differentials $$ \mathfrak{Q}(z)d{z^2} $$ 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 $$ \mathfrak{Q}(z) $$ 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.

Statistical behaviour of the leaves of Riccati foliations

AbstractWe introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one and hyperbolic, corresponding to the unique complete metric of curvature −1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ− and μ+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to μ− and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (SRB)-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.