Theory Of Riemann Surfaces Research Articles

Finite simple permutation groups acting with fixity 4

Abstract Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we classify all finite simple groups that have a faithful transitive action with fixity 4. We also give details about all possible such actions.

Some Differential Equations for the Riemann $\theta$-Function on Jacobians

We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in Arakelov theory of Riemann surfaces.

Crossing the Transcendental Divide: From Translation Surfaces to Algebraic Curves

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of L shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins–Strebel representatives, a dense family of translation surfaces that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.

Recent Developments in the Theory of Riemann Surfaces: A Comprehensive Overview

Recent Developments in the Theory of Riemann Surfaces: A Comprehensive Overview

Riemann-Roch theorem and Kodaira-Serre duality

Abstract The Riemann-Roch theorem is of utmost importance and a vital tool to the fields of complex analysis and algebraic geometry, specifically in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this paper is to give two proofs of this important theorem and explore some of its numerous consequences. As an application, we compute the genus of some interesting algebraic curves or Riemann surfaces.

Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaces

The results in this article are based on the classification of finite simple groups that act with fixity 3. Motivated by the theory of Riemann surfaces, we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case, we give detailed information about the possible branching data of the surface.

Recent Progress in Complex Analysis: From Riemann Surfaces to Holomorphic Dynamics

Complex analysis is a fundamental branch of mathematics with wide-ranging applications in various fields. This paper provides an overview of recent progress in complex analysis, focusing on two key areas: Riemann surfaces and holomorphic dynamics. We begin by discussing the historical development of complex analysis, highlighting the contributions of Cauchy, Riemann, and Weierstrass. We then delve into the theory of Riemann surfaces, including their definition, basic properties, and classification theorems. Next, we explore holomorphic dynamics, examining its definition, fundamental concepts, and recent advances. We also explore the interactions between Riemann surfaces and holomorphic dynamics, showcasing the unifying principles in complex analysis. Finally, we discuss the applications of complex analysis in quantum mechanics, number theory, and other areas of mathematics and physics. This paper aims to provide a comprehensive overview of recent developments in complex analysis and its implications for mathematics and beyond.

Electrostatics and Riemann surfaces

Using techniques from geometry and complex analysis in their simplest form, we present a derivation of electric fields on surfaces with non-trivial topology. A byproduct of this analysis is an intuitive visualization of elliptic functions when their argument is complex-valued. The underlying connections between these techniques and the theory of Riemann surfaces are also explained. Our goal is to provide students and instructors a quick reference article for an extraordinary topic that is not included in the standard books.

The genus two free boson in Arakelov geometry

Using Arakelov geometry, we compute the partition function of the noncompact free boson at genus two. We begin by compiling a list of modular invariants which appear in the Arakelov theory of Riemann surfaces. Using these quantities, we express the genus two partition function as a product of modular forms, as in the well-known genus one case. We check that our result has the expected obstruction to holomorphic factorization and behavior under degeneration.

Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quad-mesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices corresponds to the configurations of the poles and zeros (divisor) of the meromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel–Jacobi condition. Inversely, if a divisor satisfies the Abel–Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals the given one. Furthermore, if the meromorphic quartic differential is with finite trajectories, then it also induces a quad-mesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quad-mesh.Besides the theoretic proofs, the computational algorithm for verification of Abel–Jacobi condition is also explained in detail. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials.Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quad-mesh generation using algebraic geometric approach.

Frobenius-Projective Structures on Curves in Positive Characteristic

In the present paper we study Frobenius-projective structures on projective smooth curves in positive characteristic. The notion of Frobenius-projective structures may be regarded as an analogue, in positive characteristic, of the notion of complex projective structures in the classical theory of Riemann surfaces. By means of the notion of Frobenius-projective structures we obtain a relationship between a certain rational function, i.e., a pseudo-coordinate , and a certain collection of data which may be regarded as an analogue, in positive characteristic, of the notion of indigenous bundles in the classical theory of Riemann surfaces, i.e., a Frobenius-indigenous structure . As an application of this relationship, we also prove the existence of certain Frobenius-destabilized locally free coherent sheaves of rank two.

Large odd prime power order automorphism groups of algebraic curves in any characteristic

Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p≥0, and let Aut(X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by Zomorrodian for Riemann surfaces is |G|≤9(g−1) where the extremal case can only be obtained for d=3 and g≥10. We prove Zomorrodian's result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X/Z for a central subgroup Z of Aut(X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X, we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups.

The occurrence of finite simple permutation groups of fixity 2 as automorphism groups of Riemann surfaces

Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we study finite simple groups from a permutation action point of view. We classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most two fixed points on each non-regular orbit and at most four fixed points in total.

On the Oikawa and Arakawa Theorems for Graphs

The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine and their followers at the beginning of the century. This theory considers finite graphs as analogs of Riemann surfaces and branched coverings of graphs as holomorphic mappings. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of half-edges of the graph. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus g > 1 does not exceed 84(g — 1). Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a discrete analog of the E. Bujalance and G. Gromadzki theorem improving one of Arakawa’s results is obtained.

Riemann theta function solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy

The Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy associated with a 3 × 3 matrix spectral problem is proposed with the aid of Lenard recursion equations. By using the characteristic polynomial of Lax matrix for the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy, we introduce a three-sheeted Riemann surface Km−1 of arithmetic genus m−1, from which we derive the associated Baker–Akhiezer function, the meromorphic function on it. Based on the theory of Riemann surfaces, we construct the explicit Riemann theta function representations of the Baker–Akhiezer function, the meromorphic function, and solutions for the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy.

Closed extended [formula omitted]-spin theory and the Gelfand–Dickey wave function

We study a generalization of genus-zero r-spin theory in which exactly one insertion has a negative-one twist, which we refer to as the “closed extended” theory, and which is closely related to the open r-spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the genus-zero part of a special solution to the system of differential equations for the wave function of the rth Gelfand–Dickey hierarchy. This parallels an analogous result for the open r-spin generating function in the companion paper Buryak et al. (2018) to this work.

From Harmonic Maps to the Nonlinear Supersymmetric Sigma Model of Quantum Field Theory: at the Interface of Theoretical Physics, Riemannian Geometry, and Nonlinear Analysis

Harmonic maps from Riemann surfaces arise from a conformally invariant variational problem. Therefore, on one hand, they are intimately connected with moduli spaces of Riemann surfaces, and on the other hand, because the conformal group is noncompact, constitute a prototype for the formation of singularities, the so-called bubbles, in geometric analysis. In theoretical physics, they arise from the nonlinear sigma -model of quantum field theory. That model possesses a supersymmetric extension, coupling a harmonic map like field with a nonlinear spinor field. In the physical model, that spinor field is anticommuting. In this contribution, we analyze both a mathematical version with a commuting spinor field and the original supersymmetric version. Moreover, this model gives rise to a further field, a gravitino, that can be seen as the supersymmetric partner of a Riemann surface metric. Altogether, this leads to a beautiful combination of concepts from quantum field theory, structures from Riemannian geometry and Riemann surface theory, and methods of nonlinear geometric analysis.

Dynamic metaplectic spinor quantization: the projective correspondence for spectral dual pairs

From the historical perspective, the technique of X-ray computer tomography featured the predecessor of magnetic spin resonance tomography, which in fact is a non-invasive, high resolution, biomedical diagnostic scanning modality. Based on non-commutative harmonic analysis on the classical (2 + 1)-dimensional real Heisenberg unipotent Lie group $$\mathcal{N}$$ and the gradient controlled inversive and co-inversive chord-contact dynamics, framed by the coadjoint $$\mathcal{N}$$ -orbit model inside the real dual vector space $$\mathfrak {Lie}(\mathcal{N})^*$$ , the paper provides mathematical insight into the intrinsic electromagnetic quantum field and relativistic symmetries associated to the highly resolving clinical modality of magnetic spin resonance tomography by referring to the methodology of the basic control mechanisms of the projective duality correspondence for spectral dual pairs of real Lie groups. In terms of the projective manifold $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) \cong {\mathbb {P}}_3({\mathbb {C}})$$ , which is associated with the (2 + 1)-dimensional dual vector space $$\mathfrak {Lie}(\mathcal{N})^*$$ of the real Heisenberg nilpotent Lie algebra $$\mathfrak {Lie}(\mathcal{N})$$ , the smooth line bundle technique of dynamic metaplectic spinor quantization leads to the twisted action of the metaplectic Lie group $$\mathrm{Mp}(2,{\mathbb {R}}) = \widetilde{\mathrm{Sp}}(2,{\mathbb {R}})$$ . The transitive proper hyperbolic-parabolically ruled gradient action of the projective Lorentz–Mobius Lie group $$\mathrm{PSO}(1,3,{\mathbb {R}})$$ on $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ torque-records dihedrally its spectral incidence projective curve, the plane cubic $$E({\mathbb {C}}) \hookrightarrow {\mathbb {P}}_2({\mathbb {C}}) \cong \mathrm{Sym}^2\left( {\mathbb {P}}_1({\mathbb {C}})\right) $$ , on the two-dimensional pages of the open-book foliation inside the very round sphere $${\mathbb {S}}_3 \cong \mathrm{Spin}(3,{\mathbb {R}}) \cong \mathrm{SU}(2,{\mathbb {C}}) \hookrightarrow {\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ with angular momentum axis of the driving central universal Casimir metaplectic spinor. The projective correspondence for the spectral dual pair $$\left( \mathrm{Mp}(2,{\mathbb {R}}),\mathrm{PSO}(1,3,{\mathbb {R}})\right) $$ provides an efficient mathematical approach to the high resolution imaging modalities of magnetic spin resonance tomography, optical or ocular coherence tomography of neuro-ophthalmology, spin-polarized scanning tunneling microscopy, and the quantum field phenomena of the Hanbury Brown–Twiss experiment for photons and electrons. Due to the Eisenstein meromorphic calculus, the omni-directional interferometric detection of gravitational wavepackets of two polarizations is tomographically performed by the Abel–Jacobi inversion of the relativistic parabolic porism deviation of Kepplerian bifocal periodicity with its metaplectic spinor driven pair $$(\sigma ,{\bar{\sigma }})$$ of projective tangent involutions in the space $${\mathbb {P}}_{\mathbb {C}}\left( \mathfrak {Lie}(\mathcal{N})^*\right) $$ . The astrophysical emission of gravitational radiation is closely related to the concept of simply connected horned sphere which is homeomorphic to the compact base manifold of the Hopf principal circle bundle $${\mathbb {S}}_1 \hookrightarrow {\mathbb {S}}_3 {\mathop {\longrightarrow }\limits ^{\eta }} {\mathbb {S}}_2$$ . Riemann surface theory provides tomographic insight into the relativistic phenomenon of post-Kepplerian metaplectically driven spinor warping.

Distortion and topology

For a self mapping f: D→D of the unit disk in C which has finite distortion, we give a separation condition on the components of the set where the distortion is very large - say greater than a given constant - which implies that f still extends homeomorphically and quasisymmetrically to the boundary S = ∂D. Thus f shares its boundary values with a quasiconformal mapping whose distortion we explicitly estimate in terms of the data. This condition, uniformly separated in modulus, allows the set where the distortion is large to accumulate on the entire boundary S, but it does not allow a component to run out to the boundary - a necessary restriction. The lift of a Jordan domain in a Riemann surface to its universal cover D is always uniformly separated in modulus, and this allows us to apply these results in the theory of Riemann surfaces to identify an interesting link between the support of the high distortion of a map between surfaces and their geometry - again with explicit estimates. As part of our investigations, we study mappings ϕ: S → S which are the germs of a conformal mapping and give good bounds on the distortion of a quasiconformal extension of ϕ to the disk D. We then extend these results to the germs of quasisymmetric mappings. These appear of independent interest and identify new geometric invariants.

Global topology of hyperbolic components: Cantor circle case

We prove that in the moduli space Md of degree d⩾2 rational maps, any hyperbolic component in the disconnectedness locus and of Cantor circle type is a finite quotient of R4d−4−n×Tn, where n is determined by dynamics. The proof uses some ideas from Riemann surface theory (Abel's Theorem), dynamical system and algebraic topology.