Displaying similar documents to “The first eigenvalue of a Riemann surface.”

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.

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.

On commutativity and ovals for a pair of symmetries of a Riemann surface

Ewa Kozłowska-Walania (2007)

Colloquium Mathematicae

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We study the upper bounds for the total number of ovals of two symmetries of a Riemann surface of genus g, whose product has order n. We show that the natural bound coming from Bujalance, Costa, Singerman and Natanzon's original results is attained for arbitrary even n, and in case of n odd, there is a sharper bound, which is attained. We also prove that two (M-q)- and (M-q')-symmetries of a Riemann surface X of genus g commute for g ≥ q+q'+1 (by (M-q)-symmetry we understand a symmetry...

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.

Riemann mapping theorem in ℂⁿ

Krzysztof Jarosz (2012)

Annales Polonici Mathematici

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The classical Riemann Mapping Theorem states that a nontrivial simply connected domain Ω in ℂ is holomorphically homeomorphic to the open unit disc 𝔻. We also know that "similar" one-dimensional Riemann surfaces are "almost" holomorphically equivalent. We discuss the same problem concerning "similar" domains in ℂⁿ in an attempt to find a multidimensional quantitative version of the Riemann Mapping Theorem

Conformal actions with prescribed periods on Riemann surfaces

G. Gromadzki, W. Marzantowicz (2011)

Fundamenta Mathematicae

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It is a natural question what is the set of minimal periods of a holomorphic maps on a Riemann surface of negative Euler characteristic. Sierakowski studied ordinary holomorphic periods on classical Riemann surfaces. Here we study orientation reversing automorphisms acting on classical Riemann surfaces, and also automorphisms of non-orientable unbordered Klein surfaces to which, following Singerman, we shall refer to as non-orientable Riemann surfaces. We get a complete set of conditions...