Theory Of Riemann Surfaces Research Articles

Harmonic Morphisms and Hyperelliptic Graphs

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a two-edge-connected graph G that is not a cycle, there is at most one involution on G for which the quotient is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a nonconstant harmonic morphism to the graph B 2 consisting of two vertices connected by two edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

Poincaré's theorem for the modular group of real Riemann surfaces

Let Mod g denote the modular group of (closed and orientable) surfaces S of genus g. Each element [ h ] ∈ Mod g induces a symplectic automorphism H ( [ h ] ) of H 1 ( S , Z ) . Poincaré showed that H : Mod g → Sp ( 2 g , Z ) is an epimorphism. A real Riemann surface is a Riemann surface S together with an anticonformal involution σ. Let ( S , σ ) be a real Riemann surface, Homeo g σ be the group of orientation preserving homeomorphisms of S such that h ○ σ = σ ○ h and Homeo g , 0 σ be the subgroup of Homeo g σ consisting of those isotopic to the identity by an isotopy in Homeo g σ . The group Mod g σ = Homeo g σ / Homeo g , 0 σ plays the role of the modular group in the theory of real Riemann surfaces. In this work we describe the image by H of Mod g σ . Such image depends on the topological type of the involution σ.

Foci and Foliations of Real Algebraic Curves

We discuss diverse results whose common thread is the notion of focus of an algebraic curve. In a unified setting, which combines elements of projective geometry, complex analysis and Riemann surface theory, we explain the roles of ordinary and singular foci in results on numerical ranges of matrices, quadrature domains, Schwarzian reflection, and other topics. We introduce the notion of canonical foliation of a real algebraic curve, which places foci into the context of continuous families of plane curves and provides a useful method of visualization of all relevant structures in a planar graphical image.

Method of Riemann surfaces in the study of supercavitating flow around two hydrofoils in a channel

In the framework of the Tulin model of supercavitating flow, the problem of reconstructing the free surface of a channel and the shapes of the cavities behind two hydrofoils placed in an ideal fluid is solved in closed form. The conformal map that transforms a parametric plane with three cuts along the real axis into the triple-connected flow domain is found by quadratures. The use of the theory of Riemann surfaces (the Schottky doubles) enables the non-linear model problem to be reduced to two separate Riemann–Hilbert problems on a hyperelliptic surface of genus two. The solution to the first problem is a rational function with certain zeros and poles on a Riemann surface. The second problem is solved in terms of singular integrals with the Weierstrass kernel. The essential singularities of the solution at the infinite points of the surface due to a pole of the kernel are removed by solving a real analogue of the Jacobi inversion problem on the surface. The unknown parameters of the conformal map are recovered from a system of certain algebraic and transcendental equations.

Conformal Spherical Parametrization for High Genus Surfaces

Surface parameterization establishes bijective maps from a surface onto a topologically equivalent standard domain. It is well known that the spherical parameterization is limited to genus-zero surfaces. In this work, we design a new parameter domain, two-layered sphere, and present a framework for mapping high genus surfaces onto sphere. This setup allows us to trans- fer the existing applications based on general spherical parameterization to the field of high genus surfaces, such as remeshing, consistent parameterization, shape analysis, and so on. Our method is based on Riemann surface theory. We construct meromorphic functions on surfaces: for genus one surfaces, we apply Weierstrass $P$-functions; for high genus surfaces, we compute the quotient between two holomorphic one-forms. Our method of spherical parameterization is theoretically sound and practically efficient. It makes the subsequent applications on high genus surfaces very promising.

Meshless thin-shell simulation based on global conformal parameterization

This paper presents a new approach to the physically-based thin-shell simulation of point-sampled geometry via explicit, global conformal point-surface parameterization and meshless dynamics. The point-based global parameterization is founded upon the rigorous mathematics of Riemann surface theory and Hodge theory. The parameterization is globally conformal everywhere except for a minimum number of zero points. Within our parameterization framework, any well-sampled point surface is functionally equivalent to a manifold, enabling popular and powerful surface-based modeling and physically-based simulation tools to be readily adapted for point geometry processing and animation. In addition, we propose a meshless surface computational paradigm in which the partial differential equations (for dynamic physical simulation) can be applied and solved directly over point samples via Moving Least Squares (MLS) shape functions defined on the global parametric domain without explicit connectivity information. The global conformal parameterization provides a common domain to facilitate accurate meshless simulation and efficient discontinuity modeling for complex branching cracks. Through our experiments on thin-shell elastic deformation and fracture simulation, we demonstrate that our integrative method is very natural, and that it has great potential to further broaden the application scope of point-sampled geometry in graphics and relevant fields.

Counting congruence subgroups

Counting congruence subgroups

Regular Systems on Riemann Surfaces

In this paper, we consider the current status of the Riemann–Hilbert problem and problems closely related to it. In the global theory of differential equations with regular singular points, the Riemann– Hilbert problem is one of the central topics. Two-dimensional models in contemporary theoretical physics use fundamental facts of the theory of Riemann surfaces. One of the central problems of this theory is the Riemann–Hilbert problem and questions related to it, for example, the linear conjugation problem. A natural language for the investigation of the Riemann–Hilbert problem is the language of holomorphic bundles with connections on Riemann surfaces. It makes clear the relationship with the linear conjugation problem, which is usually considered for Holder-class functions. We develop a different approach; namely, we consider the Riemann–Hilbert problem for the Carleman– Bers–Vekua system, replacing the Wiener–Hopf factorization of a matrix-valued function by the Φ-factorization. To study deformations of complex structures on Riemann surfaces, we also consider the Beltrami equation as a particular case of the Carleman–Bers–Vekua equation. We also discuss the algebraic formulation of the Riemann–Hilbert problem in the framework of the differential Galois theory, where this problem is known as the inverse problem and is interesting for us because of its constructive nature.

Moduli of contact circles

We continue the study of linear families of contact forms on 3-manifolds begun in our paper `Contact geometry and complex surfaces'. The present paper introduces Teichmuller and moduli spaces for so-called taut contact circles. By constructing a developing map for taut contact circles, we show that these geometrically defined deformation spaces are equivalent to certain spaces of representations of the fundamental group in appropriate Lie groups. Furthermore, we relate these deformation spaces to `classical' Teichmuller theory of Riemann surfaces and orbifolds.

Discrete Riemann Surfaces and the Ising Model

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.

Belyi functions for Archimedean solids

The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) to Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semi-regular maps which correspond to the so-called Archimedean solids.

Ω-Admissible Theory

Arakelov and Faltings developed an admissible theory on regular arithmetic surfaces by using Arakelov canonical volume forms on the associated Riemann surfaces. Such volume forms are induced from the associated Kähler forms of the flat metric on the corresponding Jacobians. So this admissible theory is in the nature of Euclidean geometry, and hence is not quite compatible with the moduli theory of Riemann surfaces. In this paper, we develop a general admissible theory for arithmetic surfaces (associated with stable curves) with respect to any volume form. In particular, we have a theory of arithmetic surfaces in the nature of hyperbolic geometry by using hyperbolic volume forms on the associated Riemann surfaces. Our theory is proved to be useful as well: we have a very natural Weil function on the moduli space of Riemann surfaces, and show that in order to solve the arithmetic Bogomolov-Miyaoka-Yau inequality, it is sufficient to give an estimation for Petersson norms of some modular forms. 1991 Mathematics Subject Classification: 11G30, 11G99, 14H15, 53C07, 58A99.

Geometry of Riemann surfaces based on closed geodesics

The paper presents a survey on recent results on the geometry of Riemann surfaces showing that the study of closed geodesics provides a link between different aspects of Riemann surface theory such as hyperbolic geometry, topology, spectral theory, and the theory of arithmetic Fuchsian groups. Of particular interest are the systoles, the shortest closed geodesics of a surface; their study leads to the hyperbolic geometry of numbers with close analogues to classical sphere packing problems.

Generic Metrics and Connections on Spin- and Spin c -Manifolds

We study the dependence of the dimension h0(g,A) of the kernel of the Atyiah-Singer Dirac operator \({\cal D}_{g,A}\) on a spinc-manifold M on the metric g and the connection A. The main result is that in the case of spin-structures the value of h0(g) for the generic metric is given by the absolute value of the index provided \(\dim M\in\{3,4\}\). In dimension 2 the mod-2 index theorems have to be taken into a account and we obtain an extension of a classical result in the theory of Riemann surfaces. In the spinc-case we also discuss upper bounds on h0(g,A) for generic metrics, and we obtain a complete result in dimension 2. The much simpler dependence on the connection A and applications to Seiberg–Witten theory are also discussed.

P-adic theta functions and solutions of the KP hierarchy

Based on Schottky uniformization theory of Riemann surfaces, we construct a universal power series for (Riemann) theta function solutions of the KP hierarchy. Specializing this power series to the coordinates associated with Schottky groups overp-adic fields, we show that thep-adic theta functions of Mumford curves give solutions of the KP hierarchy.

UNIFORMIZATION THEORY AND 2D GRAVITY I: LIOUVILLE ACTION AND INTERSECTION NUMBERS

This is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classical Liouville action appears in the expression for the correlators of topological gravity. Next we derive an inequality involving the cutoff of 2D gravity and the background geometry. Another result, still related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. We introduce operators covariantized by means of the inverse map of uniformization. These operators have interesting properties, including holomorphicity. In particular, they are crucial for showing that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle and vanishes for deformation of the complex structure induced by the harmonic Beltrami differentials. By means of the inverse map we propose a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for the holomorphic covariant operators defining higher order cocycles and anomalies which are related to W algebras. Finally we face the problem of considering the positivity of eσ, with σ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces. These functions, whose underlying number-theoretic structure seems related to Fuchsian groups and to the eigenvalues of the Laplacian, are quite basic and may provide the building blocks for properly investigating the long-standing uniformization problem posed by Klein, Koebe and Poincaré.

Classification of Riemannian manifolds in nonlinear potential theory

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsOG⊂OHP⊂OHB⊂OHD. In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Toki. Our main objective is to show that such inclusions are proper also in the generalized case.

Riemann surfaces and the geometrization of 3-manifolds

About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of hyperbolic geometry, as natural as Euclid’s regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos], so topology and geometry mesh harmoniously in dimension 3. This remarkable theorem applies to all 3-manifolds which can be built up in an inductive way from 3-balls, i.e. Haken manifolds. Thurston’s construction of a hyperbolic structure is also inductive. At the inductive step one must find the right geometry on an open 3-manifold so that its ends may be glued together. Using quasiconformal deformations, the gluing problem can be formulated as a fixed-point problem for a map of Teichmuller space to itself. Thurston proposes to find the fixed point by iterating this map. Here we outline Thurston’s construction, and sketch a new proof that the iteration converges. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently

Anomalous BRST ward identity in string theory

BRST transformations are studied in the path integral approach to string theory of Riemann surfaces of genus h ⩾ 2. The BRST Ward identity (WI) is shown to be anomalous, the anomaly being due to non-invariance of the functional integration domain under BRST transformations. The distinction between complete Lagrange BRST transformations including the metric and the auxiliary field and the commonly used “truncated” BRST transformation is discussed in detail. The problem of decoupling of spurions from physical operators is investigated.

Three-dimensional hyperelliptic manifolds

Let X3 = H3, E3, S3, H2 × E1, S2 × E1, T1(H2), Nil of Solv be one of the eight 3-dimensional geometrics of Thurston [10] and G be a discrete group of isometrics of X3 acting without fixed points. A manifold M3 = X3/G is said to be hyperelliptic if there is an isometric involution τ on it such that the factor space M3/ is diffeomorphic to the 3-sphere S3. In analogy with the theory of Riemann surfaces we call involution.