GENUS 2 SEMI-REGULAR COVERINGS WITH LIFTING SYMMETRIES

AbstractIn this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

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Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-021-09394-2 ◽

2021 ◽

Vol 24 (3) ◽

Author(s):

Alexander I. Bobenko ◽

Ulrike Bücking

Keyword(s):

Error Estimate ◽

Branch Point ◽

Riemann Surfaces ◽

Riemann Sphere ◽

Holomorphic Functions ◽

Convergence Result ◽

Edge Length ◽

Ramified Covering ◽

Compact Riemann Surfaces ◽

Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

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On Rings with a Certain Type of Factorization and Compact Riemann Surfaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-055-7 ◽

1990 ◽

Vol 42 (6) ◽

pp. 1041-1052

Author(s):

Pascual Cutillas Ripoll

Keyword(s):

Riemann Surface ◽

Finite Subset ◽

Riemann Surfaces ◽

Compact Riemann Surface ◽

Meromorphic Functions ◽

Irreducible Elements ◽

Finite Sums ◽

Compact Riemann Surfaces

AbstractLet be a compact Riemann surface, be the complement of a nonvoid finite subset of and A() be the ring of finite sums of meromorphic functions in with finite divisor. In this paper it is proved that every nonzero f ∈ A() can be decomposed as a product αβ, where α is either a unit or a product of powers of irreducible elements of A(), uniquely determined by f up to multiplication by units, and β is a product of functions of the type eφ – 1, with φ holomorphic and nonconstant in . Furthermore, a similar result is obtained for a certain class of subrings of A().

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Conformal immersions of compact Riemann surfaces into the 2n-sphere (n ≥ 2)

Nagoya Mathematical Journal ◽

10.1017/s0027763000005535 ◽

1996 ◽

Vol 141 ◽

pp. 79-105 ◽

Cited By ~ 2

Author(s):

Jun-Ichi Hano

Keyword(s):

Riemann Surface ◽

Positive Integer ◽

Euclidean Space ◽

Unit Sphere ◽

Riemann Surfaces ◽

Compact Riemann Surface ◽

Minimal Immersion ◽

Dimensional Euclidean Space ◽

Compact Riemann Surfaces

The purpose of this article is to prove the following theorem:Let n be a positive integer larger than or equal to 2, and let S be the unit sphere in the 2n + 1 dimensional Euclidean space. Given a compact Riemann surface, we can always find a conformal and minimal immersion of the surface into S whose image is not lying in any 2n — 1 dimensional hyperplane.This is a partial generalization of the result by R. L. Bryant. In this papers, he demonstrates the existence of a conformal and minimal immersion of a compact Riemann surface into S2n, which is generically 1:1, when n = 2 ([2]) and n = 3 ([1]).

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Harmonic Morphisms as Sphere Bundles Over Compact Riemann Surfaces

International Journal of Mathematics ◽

10.1142/s0129167x97000445 ◽

1997 ◽

Vol 08 (07) ◽

pp. 935-942

Author(s):

Sigmundur Gudmundsson

Keyword(s):

Riemann Surface ◽

Homotopy Type ◽

Riemann Surfaces ◽

Compact Riemann Surface ◽

Harmonic Morphisms ◽

Sphere Bundle ◽

Harmonic Morphism ◽

Totally Geodesic ◽

Compact Riemann Surfaces

We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.

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ON THE KOSTERLITZ–THOULESS TRANSITION ON COMPACT RIEMANN SURFACES

Modern Physics Letters B ◽

10.1142/s0217984993000199 ◽

1993 ◽

Vol 07 (03) ◽

pp. 171-182 ◽

Cited By ~ 2

Author(s):

ACHILLES D. SPELIOTOPOULOS ◽

HARRY L. MORRISON

Keyword(s):

Field Theory ◽

Riemann Surface ◽

Partition Function ◽

Riemann Surfaces ◽

Compact Riemann Surface ◽

Constant Density ◽

Two Dimensional ◽

Coulombic Interaction ◽

Density Limit ◽

Compact Riemann Surfaces

A Lagrangian for the two-dimensional vortex gas is derived from a general microscopic Lagrangian for 4 He atoms on an arbitrary compact Riemann Surface without boundary. In the constant density limit the vortex Hamiltonian obtained from this Lagrangian is found to be the same as the Kosterlitz and Thouless Coulombic interaction Hamiltonian. The partition function for the Kosterlitz–Thouless ensemble on the general compact is formulated and mapped into the sine–Gordon field theory.

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