The Conformal Mapping of Simply-Connected Riemann Surfaces

1949 ◽  
Vol 50 (3) ◽  
pp. 686 ◽  
Author(s):  
Maurice Heins
Keyword(s):  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Simply Connected

scholarly journals The Conformal Mapping of Simply Connected Riemann Surfaces II

1957 ◽  
Vol 12 ◽  
pp. 139-143
Author(s):  
Maurice Heins
Keyword(s):  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Present Note ◽  
Universal Covering ◽  
Riemann Mapping ◽  
Classical Theorem ◽  
Countable Base ◽  
Riemann Mapping Theorem ◽  
Technical Interest ◽  
Simply Connected

On reviewing recently the proof which I gave for the Riemann mapping theorem for simply-connected Riemann surfaces several years ago [2], I observed that the argument which I used could be so modified that the assumption of a countable base could be completely eliminated. The problem of treating the Riemann mapping theorem without this assumption has been current for some time. The object of the present note is to give an account of a solution of this question. Of course, the classical theorem of Radó permits us to dispense with an attack on the Riemann mapping theorem which does not appeal to the countable base assumption. In this connection, we recall that Nevanlinna [4] has given a straightforward potential-theoretic treatment of the Radó theorem in which neither the Riemann mapping theorem (nor the notion of a universal covering) enters as they do in Radó’s proof. Nevertheless, a certain technical interest attaches to a direct treatment of the Riemann mapping theorem without the countable base assumption. An immediate byproduct of such a treatment is a simple proof of the Radó theorem which invokes the notion of a universal covering but in a manner different from that of Radó’s proof. Indeed, it suffices to note that a manifold has a countable base if the domain of a universal covering does.

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 Dirichlet Problems on Riemann Surfaces and Conformal Mappings

1951 ◽  
Vol 3 ◽  
pp. 91-137 ◽  
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Conformal Mappings ◽  
Dirichlet Problems ◽  
Boundary Correspondence ◽  
Abstract Riemann Surface

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.

scholarly journals Remarks on the Elliptic Case of the Mapping Theorem for Simply-Connected Riemann Surfaces

1955 ◽  
Vol 9 ◽  
pp. 17-20 ◽  
Author(s):  
Maurice Heins
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Riemann Surfaces ◽  
Simple Pole ◽  
Conformal Map ◽  
Elliptic Case ◽  
Extended Plane ◽  
Simply Connected ◽  
Conformal Equivalence

It is well-known that the conformal equivalence of a compact simply-connected Riemann surface to the extended plane is readily established once it is shown that given a local uniformizer t(p) which carries a given point p0 of the surface into 0, there exists a function u harmonic on the surface save at p0 which admits near p0 a representation of the form(α complex 0; h harmonic at p0). For the monodromy theorem then implies the existence of a meromorphic function on the surface whose real part is u. Such a meromorphic function has a simple pole at p0 and elsewhere is analytic. It defines a univalent conformal map of the surface onto the extended plane.

scholarly journals Boundary Components of Riemann Surfaces

1954 ◽  
Vol 7 ◽  
pp. 65-83
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Dirichlet Problem ◽  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Boundary Values ◽  
Abstract Riemann Surface

The boundary components of an abstract Riemann surface were defined by B. v. Kérékjértó [7] and utilized in the book [14] written by S. Stoïlow. It is the purpose of the present paper to investigate their images under conformal mapping and to solve the Dirichlet problem with boundary values distributed on them.

scholarly journals Some results on Teichmüller spaces of Klein surfaces

1997 ◽  
Vol 39 (1) ◽  
pp. 65-76
Author(s):  
Pablo Arés Gastesi
Keyword(s):  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Topological Type ◽  
Uniformization Theorem ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Simply Connected Domain ◽  
Nonorientable Surfaces ◽  
Simply Connected ◽  
Orientable Surfaces

The deformation theory of nonorientable surfaces deals with the problem of studying parameter spaces for the different dianalytic structures that a surface can have. It is an extension of the classical theory of Teichmüller spaces of Riemann surfaces, and as such, it is quite rich. In this paper we study some basic properties of the Teichmüller spaces of non-orientable surfaces, whose parallels in the orientable situation are well known. More precisely, we prove an uniformization theorem, similar to the case of Riemann surfaces, which shows that a non-orientable compact surface can be represented as the quotient of a simply connected domain of the Riemann sphere, by a discrete group of Möbius and anti-Möbius transformation (mappings whose conjugates are Mobius transformations). This uniformization result allows us to give explicit examples of Teichmüller spaces of non-orientable surfaces, as subsets of deformation spaces of orientable surfaces. We also prove two isomorphism theorems: in the first place, we show that the Teichmüller spaces of surfaces of different topological type are not, in general, equivalent. We then show that, if the topological type is preserved, but the signature changes, then the deformations spaces are isomorphic. These are generalizations of the Patterson and Bers-Greenberg theorems for Teichmüller spaces of Riemann surfaces, respectively.

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