Corrections and complements to ��Once-holed tori embedded in Riemann surfaces��

Once-holed tori are the most primitive noncompact Riemann surfaces of positive genus, and can be used to measure the sizes of handles of Riemann surfaces of positive genus. We study 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 correct some results given in Masumoto (Math. Z. 257:453?C464, 2007).     

Holomorphic mappings of once-holed tori

Let T be the space of marked once-holed tori and Y₀ be a Riemann surface with marked handle. We investigate geometric properties of the set T_a[Y₀] of X ∈ T that allow holomorphic mappings of X into Y₀. We also examine the set T_c[Y₀] of marked once-holed tori conformally embedded into Y₀. It turns out that T_a[Y₀] and T_c[Y₀] have several properties in common. Our basic tool is a new notion, called a handle condition.     

On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus

We obtain an upper bound for the number of holomorphic mappings of a genus 3 Riemann surface onto a genus 2 Riemann surface in a series of cases. In particular, we establish that the number of holomorphic mappings of an arbitrary genus 3 Riemann surface onto an arbitrary genus 2 Riemann surface is at most 48. We show that this estimate is sharp and find pairs of Riemann surfaces for which it is attained.     

Topological Loewner theory on Riemann surfaces

We prove that topological evolution families on a Riemann surface S are rather trivial unless S is conformally equivalent to the unit disc or the punctuated unit disc. We also prove that, except for the torus where there is no non-trivial continuous Loewner chain, there is a topological evolution family associated to any topological Loewner chain and, conversely, any topological evolution family comes from a topological Loewner chain on the same Riemann surface.     

Minimal surfaces in R3 with one end¶and bounded curvature

We study complete minimal surfaces M immersed in R ³, with finite topology and one end. We give conditions which oblige M to be conformally a compact Riemann surface punctured in one point, and we show that M can be parametrized by meromorphic data on this compact Riemann surface. The goal is to prove that when M is also embedded, then the end of M is asymptotic to an end of a helicoid (or M is a plane). Received: 13 January 1997 / Revised version: 15 September 1997     

Dirac eigenvalue estimates on surfaces

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or, alternatively, by stable norms of certain cohomology classes. In case of the 2-torus we obtain a positive lower bound for all Riemannian metrics and all nontrivial spin structures. For higher genus g the estimate is given by The corresponding estimate also holds for the -spectrum of the Dirac operator on a noncompact complete surface of finite area. As a corollary we get positive lower bounds on the Willmore integral for all 2-tori embedded in . Received: 15 May 2001; in final form: 11 September 2001 / Published online: 1 February 2002     

Uniqueness of the Riemann minimal examples

We prove that a properly embedded minimal surface in R ³ of genus zero with infinite symmetry group is a plane, a catenoid, a helicoid or a Riemann minimal example. We introduce the language of Hurwitz schemes to understand the underlying moduli space of surfaces in our setting. Oblatum 30-V-1997 & 5-VIII-1997     

Noncompact surfaces are packable

We show that every noncompact Riemann surface of finite type supports a circle packing. This extends earlier work of Robert Brooks [6] and Phil Bowers and Ken Stephenson [3, 4], who showed that the packable surfaces are dense in moduli space. The author gratefully acknowledges the support of the Texas Tech University Research Enhancement Fund.     

A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials

In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of uniform thickness of hyperbolic Riemann surfaces in terms of invariant metrics.     

Topology of real algebraic curves

The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.     

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.     

Fixed points of analytic self-mappings of Riemann surfaces

The existence of fixed points for an analytic self-mapping of a Riemann surface often permits strong conclusions about the mapping. For hyperbolic Riemann surfaces fixed point conditions that imply an analytic self-mapping is actually a conformal automorphism are given. For instance, an analytic self-mapping of a hyperbolic Riemann surface with two fixed points must be a conformal automorphism of finite order. On the other hand, for surfaces of finite genus estimates of the order of a conformal automorphism are obtained from fixed point information. For example, on a Riemann surface of genus g a conformal automorphism with 2g+3 fixed points is the identity.     

Uniformization of Riemann surfaces revisited

Annals of Global Analysis and Geometry - We give an elementary and self-contained proof of the uniformization theorem for noncompact simply connected Riemann surfaces.     

Classification of compact lorentzian 2-orbifolds with noncompact full isometry groups

Among closed Lorentzian surfaces, only flat tori can admit noncompact full isometry groups. Moreover, for every n ≥ 3 the standard n-dimensional flat torus equipped with canonical metric has a noncompact full isometry Lie group. We show that this fails for n = 2 and classify the flat Lorentzian metrics on the torus with a noncompact full isometry Lie group. We also prove that every two-dimensional Lorentzian orbifold is very good. This implies the existence of a unique smooth compact 2-orbifold, called the pillow, admitting Lorentzian metrics with a noncompact full isometry group. We classify the metrics of this type and make some examples.     

On Riemann surfaces with non-unique cyclic trigonal morphism

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called cyclic trigonal Riemann surface. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus greater or equal to 5. Using the characterization of cyclic trigonality by Fuchsian groups given in [3], we obtain the Riemann surfaces of low genus with non-unique trigonal morphisms. Partially supported by BFM2002-4801. Partially supported by the Swedish Research Council (VR)     

On cyclic <Emphasis Type="Italic">p</Emphasis>-gonal Riemann surfaces with several <Emphasis Type="Italic">p</Emphasis>-gonal morphisms

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.     

Symmetry types of cyclic covers of the sphere

We examine all compact Riemann surfaces of genus greater than one which admit a cyclic group of automorphisms that yields a covering of the Riemann sphere with exactly three branch points. We determine the number of non-conjugate symmetries of each of these surfaces. For each symmetry, we determine the number of ovals it fixes and whether the orbit space under the symmetry is orientable or not. This yields the species of each symmetry and the symmetry type of each surface. Explicit defining equations of each surface and symmetry are given.     

A New Proof of the Existence of Embedded Surfaces with Anosov Geodesic Flow

We give a new proof of the existence of compact surfaces embedded in ?³ with Anosov geodesic flows. This proof starts with a noncompact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.     

Abel--Jacobi Mapping for Real Hyperelliptic Riemann Surfaces

We study the degrees of the Abel--Jacobi mapping on hyperelliptic Riemann surfaces of arbitrary genus and the restrictions of the corresponding mappings to the symmetric powers of the real locus of the given Riemann surface.