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Displaying similar documents to “On commutativity and ovals for a pair of symmetries of a Riemann surface”

On pq-hyperelliptic Riemann surfaces

Ewa Tyszkowska (2005)

Colloquium Mathematicae

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A compact Riemann surface X of genus g > 1 is said to be p-hyperelliptic if X admits a conformal involution ϱ, called a p-hyperelliptic involution, for which X/ϱ is an orbifold of genus p. If in addition X admits a q-hypereliptic involution then we say that X is pq-hyperelliptic. We give a necessary and sufficient condition on p,q and g for existence of a pq-hyperelliptic Riemann surface of genus g. Moreover we give some conditions under which p- and q-hyperelliptic involutions of...

On ovals on Riemann surfaces.

Grzegorz Gromadzki (2000)

Revista Matemática Iberoamericana

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We prove that k (k ≥ 9) non-conjugate symmetries of a Riemann surface of genus g have at most 2g - 2 + 2(9 - k) ovals in total, where r is the smallest positive integer for which k ≤ 2. Furthermore we prove that for arbitrary k ≥ 9 this bound is sharp for infinitely many values of g.

Koebe's general uniformisation theorem for planar Riemann surfaces

Gollakota V. V. Hemasundar (2011)

Annales Polonici Mathematici

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We give a complete and transparent proof of Koebe's General Uniformisation Theorem that every planar Riemann surface is biholomorphic to a domain in the Riemann sphere ℂ̂, by showing that a domain with analytic boundary and at least two boundary components on a planar Riemann surface is biholomorphic to a circular-slit annulus in ℂ.

Riemann problem on the double of a multiply connected circular region

V. V. Mityushev (1997)

Annales Polonici Mathematici

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The Riemann problem has been solved in [9] for an arbitrary closed Riemann surface in terms of the principal functionals. This paper is devoted to solution of the problem only for the double of a multiply connected region and can be treated as complementary to [9,1]. We obtain a complete solution of the Riemann problem in that particular case. The solution is given in analytic form by a Poincaré series.

Non-maximal cyclic group actions on compact Riemann surfaces.

David Singerman, Paul Watson (1997)

Revista Matemática de la Universidad Complutense de Madrid

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We say that a finite group G of automorphisms of a Riemann surface X is non-maximal in genus g if (i) G acts as a group of automorphisms of some compact Riemann surface Xg of genus g and (ii), for all such surfaces Xg , |Aut Xg| > |G|. In this paper we investigate the case where G is a cyclic group Cn of order n. If Cn acts on only finitely many surfaces of genus g, then we completely solve the problem of finding all such pairs (n,g).