scholarly journals Meromorphic functions on a compact Riemann surface and associated complete minimal surfaces

1989 ◽  
Vol 105 (3) ◽  
pp. 706-706 ◽  
Author(s):  
Kichoon Yang
Keyword(s):  
Riemann Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Meromorphic Functions

scholarly journals Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

Hermite-Padé approximants for meromorphic functions on a compact Riemann surface

2017 ◽  
Vol 72 (4) ◽  
pp. 671-706 ◽  
Author(s):  
A V Komlov ◽  
R V Palvelev ◽  
S P Suetin ◽  
E M Chirka
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Meromorphic Functions ◽  
Padé Approximants ◽  
Pade Approximants

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

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().

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.

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