scholarly journals On s-extremal Riemann surfaces of even genus

2020 ◽  
Author(s):  
Ewa Kozłowska-Walania
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Minimal Genus

AbstractWe consider Riemann surfaces of even genus g with the action of the group $$\mathcal {D}_n\times \mathbb {Z}_2$$ D n × Z 2 , with n even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called s-extremal. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an s-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called extremal. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not.

Invariant Subspaces on Riemann Surfaces

1966 ◽  
Vol 18 ◽  
pp. 399-403 ◽  
Author(s):  
Michael Voichick
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Finite Number ◽  
Harmonic Measure ◽  
Riemann Surfaces ◽  
Invariant Subspaces ◽  
Riemann Surface With Boundary ◽  
Analytic Curves

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.

ALTERNATING GROUPS AS AUTOMORPHISM GROUPS OF RIEMANN SURFACES

2006 ◽  
Vol 16 (01) ◽  
pp. 91-98
Author(s):  
J. J. ETAYO GORDEJUELA ◽  
E. MARTÍNEZ
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Upper Bound ◽  
Riemann Surfaces ◽  
Automorphism Groups ◽  
Alternating Groups ◽  
Minimal Genus

In this work we give pairs of generators (x, y) for the alternating groups An, 5 ≤ n ≤ 19, such that they determine the minimal genus of a Riemann surface on which An acts as the automorphism group. Using these results we prove that A15 is the unique of these groups that is an H*-group, i.e., the groups achieving the upper bound of the order of an automorphism group acting on non-orientable unbordered surfaces.

scholarly journals Behaviour of Green Lines at Royden’s Boundary of Riemann Surfaces

1964 ◽  
Vol 24 ◽  
pp. 1-27 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Green Function ◽  
Riemann Surfaces ◽  
Fixed Pole ◽  
The Green Function

The aim of this paper is to investigate the behaviour of Green lines at Royden’s boundary Γ of a Riemann surface R with the Green function g(z, o) with the fixed pole o in R. We denote by the totality of Green lines L issuing from the fixed point o.

scholarly journals RIEMANN SURFACES AND VERTEX OPERATOR ALGEBRAS

2012 ◽  
Vol 09 (08) ◽  
pp. 1250063
Author(s):  
K. M. BUGAJSKA
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Riemann Surfaces ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Operator Algebras

We show that for any fixed point P0 on a Riemann surface Σ the distinct realizations of cocycles in [Formula: see text] correspond to the natural appearances of the standard Heisenberg vertex operator algebra Π(P0) and to the commutative Heisenberg vertex operator algebra Π0(P0), respectively.

scholarly journals On real trigonal Riemann surfaces

2006 ◽  
Vol 98 (1) ◽  
pp. 53 ◽  
Author(s):  
Antonio F. Costa ◽  
Milagros Izquierdo
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Connected Components ◽  
Klein Surface ◽  
Fixed Point Set ◽  
Regular Covering ◽  
Closed Riemann Surface ◽  
Point Set

A closed Riemann surface $X$ 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. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) $\sigma$ of $X$ commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry $\sigma $ is the number of connected components of the fixed point set $\mathrm{Fix}(\sigma)$ and the orientability of the Klein surface $X/\langle\sigma\rangle$. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

scholarly journals SUPERGRAVITY DESCRIPTION OF FIELD THEORIES ON CURVED MANIFOLDS AND A NO GO THEOREM

2001 ◽  
Vol 16 (05) ◽  
pp. 822-855 ◽  
Author(s):  
JUAN MALDACENA ◽  
CARLOS NUÑEZ
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Field Theories ◽  
De Sitter ◽  
Curved Manifolds ◽  
Conformal Field ◽  
Low Energies ◽  
M Theory

In the first part of this paper we find supergravity solutions corresponding to branes on worldvolumes of the form Rd×Σ where Σ is a Riemann surface. These theories arise when we wrap branes on holomorphic Riemann surfaces inside K3 or CY manifolds. In some cases the theory at low energies is a conformal field theory with two less dimensions. We find some non-singular supersymmetric compactifications of M-theory down to AdS5. We also propose a criterion for permissible singularities in supergravity solutions. In the second part of this paper, which can be read independently of the first, we show that there are no non-singular Randall-Sundrum or de-Sitter compactifications for large class of gravity theories.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

Rings of Meromorphic Functions on Non-Compact Riemann Surfaces

1969 ◽  
Vol 21 ◽  
pp. 284-300 ◽  
Author(s):  
James Kelleher
Keyword(s):  
Riemann Surface ◽  
Algebraic Structure ◽  
Riemann Surfaces ◽  
Meromorphic Functions ◽  
Doctoral Dissertation ◽  
University Of Illinois ◽  
Compact Riemann Surfaces ◽  
The University ◽  
Complex Valued

In this paper we shall be concerned with the algebraic structure of certain rings of functions meromorphic on a non-compact (connected) Riemann surface Ω. In this setting, A = A(Ω) and K= K(Ω) denote (respectively) the ring of all complex-valued functions analytic on Ω and its field of quotients, the field of functions meromorphic on Ω. The rings considered here are those subrings of K containing A,which we term A-rings of K. Most of the results given here were previously announced without proof (15) and are contained in the author's doctoral dissertation (16), completed at the University of Illinois under the direction of Professor M. Heins, whose encouragement and advice are gratefully acknowledged.

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