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Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmuller spaces. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions, Comment: arXiv admin note: text overlap with arXiv:2003.14214, arXiv:1908.05183

In his Gottingen dissertation of 1851 Bernard Riemann enunciated (p. 40 of his Werke) and attempted to prove the famous theorem that now bears his name: Theorem 9.1 Every simply-connected region other than ℂ itself is conformal to an open disk.

We show that the features of Teichmuller spaces give rise to a completely new powerful tool for solving the classical coefficient problems in geometric function theory. Here we concern the univalent functions. Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $$J(f) = J(a_{m_1}, a_{m_2},\dots , a_{m_n}) $$ on some classes of univalent functions in the unit disk naturally connected with the canonical class S. The given functional J is lifted to the Teichmuller space $$\mathbf{T}_1$$ of the punctured disk $$\mathbb {D}_{*} = \{0< |z| < 1\}$$ which is biholomorphically equivalent to the Bers fiber space over the universal Teichmuller space. This generates a positive subharmonic function on the disk $$\{|t| < 4\}$$ with $$\sup _{|t

In this paper we study harmonic maps from super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field D . In both cases, we develop the theory of Gaus transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a definition of finite type harmonic maps for a special class of maps into C P n | n + 1 and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in D P 2 whose underlying map is not harmonic.

The criteria for continuous and homeomorphic extensions to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends by Carath´eodory are proved.

We describe the limit surfaces of some equisymmetric $1$-complex dimensional families of Riemann surfaces in the boundary of the Deligne-Mumford compactification of moduli space. We provide a description of such nodal Riemann surfaces in terms of the group of automorphisms defining the family. We apply our method to some known examples.

We suggest an approximate method to find an elliptic function uniformizing a compact Riemann surface of genus 1 which is given as a ramified covering of the Riemann sphere. The method is based on including the surface into a smooth one-parametric family. We deduce a system of ordinary differential equations for critical points of elliptic functions uniformizing surfaces of the family.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian integrals. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann-Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann-Roch Theorem, and a discrete Abel-Jacobi map., Comment: 35 pages, 9 figures. Minor corrections, new Figures 3, 5, 6, and 7

The original proof-argument of Riemann in 1851 for the Riemann mapping theorem, one of the most central theorems in Complex analysis, was found faulty and essentially buried underneath the proof by $Carath{\acute{e}}odory$ of 1929, now accepted as the "textbook" proof. On the other hand,...

Given a function f : E → C ‾ from a closed subset of a Riemann surface R to the Riemann sphere C ‾ , we seek to approximate f in the spherical distance by functions meromorphic on R. As a consequence we generalize a recent extension of Mergelyan's theorem, due to Fragoulopoulou, Nestoridis and Papadoperakis [12] . The problem of approximating by meromorphic functions pole-free on E is equivalent to that of approximating by meromorphic functions zero-free on E, which in turn is related to Voronin's spectacular universality theorem for the Riemann zeta-function.

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S3 onto a genus 2 Riemann surface S2 by Hol(S3, S2). Call two mappings f and g in Hol(S3, S2) equivalent whenever there exist conformal automorphisms α and β of S3 and S2 respectively with f ◦ α = β ◦ g. It is known that Hol(S3, S2) always consists of at most two equivalence classes.

We solve the Riemann boundary value problem for holomorphic functions on a periodic non-rectifiable curve. Probably, this is the first study of the Riemann problem on infinite curve, which is not rectifiable. The solution is obtained in terms of generalized integrals over non-rectifiable path.

Several sufficient conditions for the removability of singular sets with area zero are obtained for holomorphic functions in plane domains and on Riemann surfaces, under certain metric and topological assumptions about cluster sets of these functions and the limit sets of their graphs. Bibliography: 14 titles.

We give new proofs of effective versions of the Riemann mapping theorem, its extension to multiply connected domains and the uniformization on Riemann surfaces. Astonishingly, in the presented proofs, we need barely more than computational compactness and the classical results.

In this paper, we construct a uniform formula for the Riemann solutions of the simplified Chapman-Jouguet model. Firstly, we define a new functional, and then, we obtain that the Riemann solutions can be expressed by the maximum value point of this functional, while Riemann solutions may contain some of strong detonation waves, Chapman-Jouguet detonation waves and contact discontinuities. Finally, Chapman-Jouguet deflagration waves are also discussed.

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.

In this article, a new canonical function has been established to deal with Riemann boundary-value problem of periodic analytic functions discussed in [16]. In comparison with the corresponding result in [16], the expression of solution obtained here is much simpler. Then, we demonstrate the equivalence of solutions for the homogeneous Riemann problem. What's more, we obtain the precise rank of matrix of coefficients for the system of linear algebraic equations (4.35) in [16]. Those results can simplify the discussion of Riemann problem of single-periodic polyanalytic functions in [16].

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.

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.

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.

In the moduli space Mg of Riemann surfaces of genus g ≥ 2 there is important, so-called, real locus R_g, consisting of points representing Riemann surfaces having symmetries, by which we understand antiholomorphic involutions. R_g itself is covered by the strata R^k_g, each being formed by the points representing surfaces having a symmetry of given topological type k. These strata are known to be real analytic varieties of dimension 3(g-1). Also, their topological structure is pretty well known. Natanzon-Seppälä have realized that they are connected orbifolds homeomorphic to the factors of R^(3g-3) with respect to actions of some discrete groups and Goulden–Jackson–Harer and Harer–Zagier have found their Euler characteristics, expressing them through the Riemann zeta function. However, topological properties of the whole real locus R_g were less studied. The most known fact is its connectivity, proved independently by Buser-Seppälä-Silhol, Frediani and Costa–Izquierdo. This paper can be seen as a further contribution to the study of topology of R_g, which was possible through the notion of the nerve N_g, associated to R_g and called the real nerve. We find upper bounds for its geometrical and homological dimensions and we show their sharpness for infinitely many values of odd g. Precise values of these dimensions for even g have been found by the authors in an earlier paper.

We study the relation between the holomorphic approximation property and the strong disk property for an open set of an open Riemann surface or a Stein space of pure dimension 1.

The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.

Our purpose is to define composition operators acting upon Hardy spaces of Riemann surfaces. In terms of counting functions related to analytic self-map on Riemann surfaces, the boundedness and compactness are characterized.

Integral equalities involving integrals of the logarithm of the Riemann ς-function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp(−2πt). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes.

It is shown, that the mapping class group of a surface of the genus g > 1 admits a faithful representation into the matrix group GL (6g-6, Z). The proof is based on a categorical correspondence between the Riemann surfaces and the so-called toric AF-algebras.

We define a kind of moduli space of nested surfaces and mappings, which we call a comparison moduli space. We review examples of such spaces in geometric function theory and modern Teichmuller theory, and illustrate how a wide range of phenomena in complex analysis are captured by this notion of moduli space. The paper includes a list of open problems in classical and modern function theory and Teichmuller theory ranging from general theoretical questions to specific technical problems.

We shall prove a Poincare–Bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact Riemann surface. We shall also briefly discuss the case of non-compact Riemann surfaces, and study in detail the geodesics for a holomorphic connection on a complex torus.

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.

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.

Natural Riemann extensions are pseudo-Riemannian metrics (introduced by Sekizawa and studied then by Kowalski–Sekizawa), which generalize the classical Riemann extension defined by Patterson–Walker. Let \(M\) be a manifold with an affine connection and let \(T^{*}M\) be the total space of its cotangent bundle. On \(T^{*}M\) endowed with a natural Riemann extension, we study here the Laplacian and give necessary and sufficient conditions for the harmonicity of a certain family of (local) functions. We also prove a gradient formula for natural Riemann extensions.

We prove a discrete version of the Mishou theorem on the approximation of every pair of analytic functions by shifts of the Riemann zeta-function and Hurwitz zeta-functions. For this, a certain independence hypothesis is applied.

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.

We give a simplified definition of the new version of circular symmetrization which has previously been suggested by the author for solving extremal problems in geometric function theory. A proof of the symmetrization principle for the capacities of condensers on Riemann surfaces is presented. In addition, the class of condensers under consideration is extended and all the cases of equality in the symmetrization principle are found. Bibliography: 22 titles.

In this paper, we present and study a kind of Riemann boundary value problem of non-normal type for analytic functions on two parallel curves. Making use of the method of complex functions, we give the method for solving this kind of doubly periodic Riemann boundary value problem of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.

For a function defined on an arbitrary subset of a Riemann surface, we give conditions which allow the function to be extended conformally. One folkloric consequence is that two common definitions of an analytic arc in $${\mathbb{C}}$$ are equivalent.

The existence of an explicit and canonical cell decomposition of the moduli space of closed Riemann surfaces of genus two shows that each Riemann surface of genus two can be parametrised by a 12-tuple of real numbers which corresponds to the angle coordinates of a graph associated to the surface. This suggests a Circle Pattern Uniformization Problem that we have defined and solved for three classical Riemann surfaces of genus two. Although in general, finding the exact algebraic equations corresponding to a hyperbolic surface from angle coordinates is a hard problem, we prove that known numerical methods can be applied to find approximated equations of Riemann surfaces of genus two from their angle coordinates and graph data for a large family of Riemann surfaces of genus two.

We discuss the value distribution of Borel measurable maps which are holomorphic along leaves of complex laminations. In the case of complex lamination by hyperbolic Riemann surfaces with an ergodic harmonic measure, we have a defect relation appearing in Nevanlinna theory. It gives a bound of the number of omitted hyperplanes in general position by those maps.

The Riemann problem for a special Keyfitz-Kranzer system is investigated and then seven different Riemann solutions are constructed. When the initial data are chosen as three piecewise constant states under suitable assumptions, the global solutions to the perturbed Riemann problem are constructed explicitly by studying all occurring wave interactions in detail. Furthermore, the stabilities of solutions are obtained under the specific small perturbations of Riemann initial data.

Preface.- Introduction.- 1.Athanase Papadopoulos: Looking backward: From Euler to Riemann .- 2.Jeremey Gray: Riemann on geometry, physics, and philosophy - some remarks.- 3.Hubert Goenner: Some remarks on a contribution to electrodynamics by Bernhard Riemann.- 4.Christian Houzel: Riemann's Memoir UEber das Verschwinden der Theta-Functionen.- 5.Sumio Yamada: Riemann's work on minimal surfaces.- 6. Athanase Papadopoulos: Physics in Riemann's mathematical papers.- 7.Athanase Papadopoulos: Cauchy and Puiseux: Two precursors of Riemann.- 8.Athanase Papadopoulos: Riemann surfaces: Reception by the French school.- 9.Ken'ichi Ohshika: The origin of the notion of manifold: from Riemann's Habilitationsvortrag onward.- 10.Franck Jedrzejewski: Deleuze et la geometrie riemannienne : une topologie des multiplicites.- 11.Arkady Plotnitsky: Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts.- 12.Feng Luo: The Riemann mapping theorem and its discrete counterparts.- 13.Norbert A'Campo, Vincent Alberge and Elena Frenkel: The Riemann-Roch theorem.- 14.Victor Pambuccian, Horst Struve and Rolf Struve: Metric geometries in an axiomatic perspective.- 15.Toshikazu Sunada: Generalized Riemann sums.- 16.Jacques Franchi: From Riemannian to Relativistic Diffusions.- 17.Andreas Hermann and Emmanuel Humbert: On the Positive Mass Theorem for closed Riemannian manifolds.- 18.Marc Mars: On local characterization results in geometry and gravitation.- 19.Jean-Philippe Nicolas: The conformal approach to asymptotic analysis.- 20.Lizhen Ji: Bernhard Riemann and his work.

Three months before his death in 1866, Riemann left a set of notes to K. Hattendorff, a disciple of his, on minimal surfaces with boundary. Afterwards, Hattendorff supplied the text to the notes mostly consisting of computations, which became the two papers on the subject: “On the surface of least area with a given boundary” and “Examples of surfaces of least area with a given boundary.” We will go over the expositions and provide an overview from the modern viewpoint, make some comments on Riemann-Hattendorff’s text, and compare the work with that of Weierstrass on the same subject.

We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann’s predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler.

The theory of Teichmuller spaces is a classical topic in the complex analysis of Riemann surfaces. In the case of the unit disk, Teichmuller spaces have their origins in geometric function theory. In particular, the Bloch space and the space BMOA play important roles in the theory of Teichmuller spaces. The purpose of this chapter is to generalize some of the geometric function theory of Teichmuller spaces to the context of \(\mathcal{Q}_{K}\) spaces.