Congruence subgroups and maximal Riemann surfaces

A global maximal Riemann surface is a surface of constant curvature ?1 with the property that the length of its shortest simple closed geodesic is maximal with respect to all surfaces of the corresponding Teichmüller space. I show that the Riemann surfaces that correspond to the principal congruence subgroups of the modular group are global maximal surfaces. This result provides a strong geometrical reason that the Selberg conjecture, which says that these surfaces have no eigenvalues of the Laplacian in the open interval (0, 1/4), is true.     

Canonical geometrically ruled surfaces

We prove the existence of canonical scrolls; that is, scrolls playing the role of canonical curves. First of all, they provide the geometrical version of Riemann Roch Theorem: any special scroll is the projection of a canonical scroll and they allow to understand the classification of special scrolls in P^N. Canonical scrolls correspond to the projective model of canonical geometrically ruled surfaces over a smooth curve. We also prove that the generic canonical scroll is projectively normal except in the hyperelliptic case and for very particular cases in the nonhyperelliptic situation. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Crystallography and Riemann surfaces

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ² and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ² generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut, form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior with respect to the complex parameter which specifies the cut.     

Once-holed tori embedded in Riemann surfaces

Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and consitute a partially ordered set, the order being defined in terms of conforaml embeddings. We consider some families of once-holed tori that are conformally embedded in target Riemann surfaces of conformal mappings of a given noncompact Riemann surface of genus one, and establish an analogue of the one-quarter theorem of Koebe. We also investigate families of once-holed tori conformally embedded in a Riemann surface of positive genus.      

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ?×?^?, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ?×?^? of recent results of Wold and Forstneri? on the long-standing problem of properly embedding open Riemann surfaces into ?², with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.     

Multiplicative Weierstrass Points and the Jacobi Variety of a Compact Riemann Surface

In the previous papers of the present author, a general theory of multiplicative Weierstrass points on compact Riemann surfaces for arbitrary characters was developed. In the present paper, some additional relations between multiplicative Weierstrass points on a compact Riemann surface for an arbitrary character and special subsets in the Jacobi variety, the canonical embedding of a compact Riemann surface into a projective space, are established. We not distinction between classical Weierstrass points and multiplicative Weierstrass points on a compact Riemann surface.     

On some generalizations of the classical Riemann problem

The special class of the homogeneous vector boundary Riemann problems on the finite sequence of algebraic surfaces is investigated completely. Its coefficients are the noncommutative permutative matrices of the arbitrary but not prime order, and boundary conditions are given on the system of open contours. The constructive solution procedure and definite structure of the canonical solution matrix are obtained and present some generalizations of the classical Riemann problem. Simultaneously the corresponding class of algebraic equations for the appropriate covering surfaces is formed explicitly too. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topologically unique maximal elementary abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ≥2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.     

Asymmetric and pseudo-symmetric hyperelliptic surfaces

 In this paper we examine hyperelliptic Riemann surfaces which possess an anticonformal automorphism but are not symmetric. We determine that all such surfaces must have a full automorphism group which is either cyclic, or an abelian extension of a cyclic group by ℤ₂. We give defining equations for all of these hyperelliptic surfaces and show how they can be constructed by using NEC groups. As a special case, we determine all hyperelliptic surfaces which are pseudo-symmetric but not symmetric. Received: 15 November 2001     

Special discontinuities in nonlinearly elastic media

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.     

On new kinds of Teichmüller spaces

We introduce a Teichmüller space for a Riemann surface withn distinguished points. Ifn=0 this is the ordinary Teichmüller space. Forn=1, in special cases, it represents the Teichmüller curve and the universal covering space of the Teichmüller curve. The corresponding modular groups and Riemann spaces are investigated. Some purely topological applications on homotopy of self-maps of surfaces are obtained. Research partially supported by NSF Grant GP-19572. The author is currently a Guggenheim Memorial Fellow.     

{Omega}-Admissible Theory

Arakelov and Faltings developed an admissible theory on regulararithmetic surfaces by using Arakelov canonical volume formson the associated Riemann surfaces. Such volume forms are inducedfrom the associated Kähler forms of the flat metric onthe corresponding Jacobians. So this admissible theory is inthe nature of Euclidean geometry, and hence is not quite compatiblewith the moduli theory of Riemann surfaces. In this paper, wedevelop 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 thenature of hyperbolic geometry by using hyperbolic volume formson the associated Riemann surfaces. Our theory is proved tobe useful as well: we have a very natural Weil function on themoduli space of Riemann surfaces, and show that in order tosolve the arithmetic Bogomolov-Miyaoka-Yau inequality, it issufficient to give an estimation for Petersson norms of somemodular forms. 1991 Mathematics Subject Classification: 11G30,11G99, 14H15, 53C07, 58A99.     

A special case of de Branges' theorem on the inverse monodromy problem

We give a new proof of a special case of de Branges' theorem on the inverse monodromy problem: when an associated Riemann surface is of Widom type with Direct Cauchy Theorem. The proof is based on our previous result (with M.Sodin) on infinite dimensional Jacobi inversion and on Levin's uniqueness theorem for conformal maps onto comb-like domains. Although in this way we can not prove de Branges' Theorem in full generality, our proof is rather constructive and may lead to a multi-dimensional generalization. It could also shed light on the structure of invariant subspaces of Hardy spaces on Riemann surfaces of infinite genus.This work was supported by the Austrian Founds zur Förderung der wissenschaftlichen Forschung, project-number P12985-TEC     

Description of π-partition of a diffeomorphism with invariant measure

For a diffeomorphism of a smooth compact Riemann manifold, retaining a measure equivalent to Riemann volume, a special invariant partition is constructed on a set where at least one value of the characteristic Lyapunov indicators is nonzero. This partition possesses properties analogous to the properties of partition into global condensing sheets for Y-diffeomorphisms while, as the complement to this set, there is partition into points. It is proven that the measurable hull of this partition coincides with the π-partition of a diffeomorphism.     

The space of harmonic Prym differentials on a variable compact Riemann surface

The harmonic Prym differentials and their period classes play an important role in the modern theory of functions on compact Riemann surfaces [1–7]. We study the harmonic Prym bundle, whose fibers are the spaces of harmonic Prym differentials on variable compact Riemann surfaces and find its connection with Gunning’s cohomological bundle over the Teichmüller space for two important subgroups of the inessential and normalized characters on a compact Riemann surface. We study the periods of holomorphic Prym differentials for essential characters on variable compact Riemann surfaces.     

On the connectedness of the set of Riemann surfaces with real moduli

The moduli space \({\mathcal {M}}_{g}\), of genus \(g\ge 2\) closed Riemann surfaces, is a complex orbifold of dimension \(3(g-1)\) which carries a natural real structure, i.e. it admits an anti-holomorphic involution \(\sigma \). The involution \(\sigma \) maps each point corresponding to a Riemann surface S to its complex conjugate \(\overline{S}\). The fixed point set of \(\sigma \) consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside \(\mathrm {Fix}(\sigma )\) is the locus \({\mathcal {M}}_{g}(\mathbb {R})\), the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppälä, and R. Silhol. The complement \(\mathrm {Fix}(\sigma )-{\mathcal {M}}_{g}(\mathbb {R})\) consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that \(\mathrm {Fix}(\sigma )\) is connected.     

Minimal genus problem for pseudo-real Riemann surfaces

A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.     

Transplantation and Jacobians of Sunada isospectral Riemann surfaces

We consider relations among the Jacobians of isospectral compact Riemann surfaces constructed using Sunada's theorem. We use a simple algebraic formulation of “transplantation” of holomorphic 1-forms and singular 1-cycles to obtain two main results. First, we obtain a geometric proof of a result of Prasad and Rajan that Sunada isospectral Riemann surfaces have isogenous Jacobians. Second, we determine a relationship (weaker than isogeny) that holds among the Jacobians of Sunada isospectral Riemann surfaces when the Jacobians’ extra structure as principally polarized abelian varieties is taken into account. We also show all Sunada isospectral manifolds have isomorphic real cohomology algebras. Finally, we exhibit transplantation of cycles explicitly in a concrete example of a pair of isospectral Riemann surfaces constructed by Brooks and Tse.     

Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node

In this article, we consider a family of compact Riemann surfaces of genus q 2 degenerating to a Riemann surface with a separating node and many non-separating nodes. We obtain the asymptotic behavior of Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces.