scholarly journals Weil–Petersson class non-overlapping mappings into a Riemann surface

2016 ◽  
Vol 18 (04) ◽  
pp. 1550060 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Unit Disk ◽  
Bergman Space ◽  
Compact Riemann Surface ◽  
Conformal Mappings ◽  
Hilbert Manifold ◽  
Genus Formula

For a compact Riemann surface of genus [Formula: see text] with [Formula: see text] punctures, consider the class of [Formula: see text]-tuples of conformal mappings [Formula: see text] of the unit disk each taking [Formula: see text] to a puncture. Assume further that (1) these maps are quasiconformally extendible to [Formula: see text], (2) the pre-Schwarzian of each [Formula: see text] is in the Bergman space, and (3) the images of the closures of the disk do not intersect. We show that the class of such non-overlapping mappings is a complex Hilbert manifold.

An Elementary Proof of a Fixed Point Theorem of J. Lewittes and D. L. McQuillan

1978 ◽  
Vol 21 (1) ◽  
pp. 99-101 ◽  
Author(s):  
Arthur K. Wayman
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Fixed Point Theorem ◽  
Compact Riemann Surface ◽  
Elementary Proof ◽  
Algebraic Function ◽  
Weierstrass Points ◽  
Algebraic Function Fields ◽  
Genus Formula ◽  
Tame Ramification

In (3), J. Lewittes establishes a connection between the number of fixed points of an automorphism of a compact Riemann surface and Weierstrass points on the surface; Lewittes′ techniques are analytic in nature. In (4), D. L. McQuillan proved the result by purely algebraic methods and extended it to arbitrary algebraic function fields in one variable over algebraically closed ground fields, but with restriction to tamely ramified places. In this paper we will give a different proof of the theorem and show that it is an elementary consequence of the Riemann-Hurwitz relative genus formula. Moreover, we can remove the tame ramification restriction.

scholarly journals Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020 ◽  
Author(s):  
Eric Schippers ◽  
Mohammad Shirazi ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Bergman Space ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Finite Collection ◽  
Approximation Theorems ◽  
Arbitrary Genus ◽  
Simply Connected

Abstract We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

scholarly journals Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650025 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Tangent Space ◽  
Riemann Surfaces ◽  
Tangent Vector ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Holomorphic Curve ◽  
Hilbert Manifold ◽  
Direct Sum ◽  
Genus Formula

Consider a Riemann surface of genus [Formula: see text] bordered by [Formula: see text] curves homeomorphic to the unit circle, and assume that [Formula: see text]. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in [Formula: see text] and [Formula: see text], and furthermore that the space of [Formula: see text] differentials in [Formula: see text] decomposes as a direct sum of infinitesimally trivial differentials and [Formula: see text] harmonic [Formula: see text] differentials. Thus the tangent space of this Teichmüller space is given by [Formula: see text] harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of [Formula: see text] harmonic Beltrami differentials.

On (q,n)-gonal pseudo-real Riemann surfaces

2017 ◽  
Vol 28 (13) ◽  
pp. 1750095 ◽  
Author(s):  
Ewa Tyszkowska
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Prime Order ◽  
Compact Riemann Surface ◽  
Quotient Spaces ◽  
Real Riemann Surfaces ◽  
Genus Formula

A compact Riemann surface [Formula: see text] of genus [Formula: see text] is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real [Formula: see text]-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order [Formula: see text] such that the quotient spaces have genus [Formula: see text].

PARABOLIC GENERALIZED MODULAR FORMS AND THEIR CHARACTERS

2009 ◽  
Vol 05 (05) ◽  
pp. 845-857 ◽  
Author(s):  
MARVIN KNOPP ◽  
GEOFFREY MASON
Keyword(s):  
Riemann Surface ◽  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Eichler Cohomology

We make a detailed study of the generalized modular forms of weight zero and their associated multiplier systems (characters) on an arbitrary subgroup Γ of finite index in the modular group. Among other things, we show that every generalized divisor on the compact Riemann surface associated to Γ is the divisor of a modular form (with unitary character) which is unique up to scalars. This extends a result of Petersson, and has applications to the Eichler cohomology.

A Note on Weierstrass Points

1967 ◽  
Vol 19 ◽  
pp. 268-272 ◽  
Author(s):  
Donald L. McQuillan
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Function Fields ◽  
Algebraic Function ◽  
Modular Functions ◽  
Weierstrass Points ◽  
Algebraic Function Fields

In (4) G. Lewittes proved some theorems connecting automorphisms of a compact Riemann surface with the Weierstrass points of the surface, and in (5) he applied these results to elliptic modular functions. We refer the reader to these papers for definitions and details. It is our purpose in this note to point out that these results are of a purely algebraic nature, valid in arbitrary algebraic function fields of one variable over algebraically closed ground fields (with an obvious restriction on the characteristic). We shall also make use of the calculation carried out in (5) to obtain a rather easy extension of a theorem proved in (6, p. 312).

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