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Dissertations / Theses on the topic 'Harmonic functions on Riemann surfaces'
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1
Vanni, Ismaele. "Meromorphic functions on compact Riemann surfaces." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2018.
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Nella presente trattazione si affronterà, utilizzando tecniche analitiche e forme differenziali, la questione dell'esistenza di funzioni meromorfe su una superficie di Riemann compatta: dopo un esame di alcune proprietà generali delle superfici di Riemann, si studieranno alcuni dei legami fra analisi e topologia di una superficie di Riemann, e successivamente da un teorema sulla risolubilità dell'equazione di Poisson su una tale superficie compatta si dedurranno una versione ristretta del teorema di Riemann-Roch e il teorema di Abel-Jacobi.
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Valli, Giorgio. "Some aspects of the theory of harmonic gauges over Riemann surfaces." Thesis, University of Warwick, 1988. http://wrap.warwick.ac.uk/109833/.
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The main subject of this Thesis is the study of harmonic maps from compact Riemann surfaces into unitary groups, and various generalisations and related subjects. Harmonic maps are critical points of the energy functional. In the case we are considering, the associated Euler-Lagrange equations are particularly simple, because of the conformal invariance of the energy for maps from surfaces, which emphasizes the role of the complex structure, and of the simplicity of the target manifold. Another important feature is that the non-linear equations are representable as O-curvature conditions for families (loops) of connections. This is the elliptic version of a phenomenon which is typical of a class of evolution equations, where it induces soliton behaviour, and complete integrability. ln this elliptic situation, this representation (due to Zakharov et al.) allows us to substitute algebraic geometry to analysis, in the description of the solutions.
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Aryasomayajula, Naga Venkata Anilatmaja. "Bounds for Green's functions on hyperbolic Riemann surfaces of finite volume." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16828.
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Im Jahr 2006, in einem Papier in Compositio Titel "Bounds auf kanonische Green-Funktionen" J. Jorgenson und J. Kramer, haben optimale Schranken für den hyperbolischen und kanonischen Green-Funktionen auf einem kompakten hyperbolischen Riemannschen Fläche definiert abgeleitet. Diese Schätzungen wurden im Hinblick auf abgeleitete Invarianten aus hyperbolischen Geometrie der Riemannschen Fläche. Als Anwendung abgeleitet sie Schranken für die kanonische Green-Funktionen durch Abdeckungen und für Familien von Modulkurven. In dieser Arbeit erweitern wir ihre Methoden nichtkompakten hyperbolischen Riemann Oberflächen und leiten ähnliche Schranken für den hyperbolischen und kanonischen Green-Funktionen auf einem nichtkompakten hyperbolischen Riemannschen Fläche definiert.In 2006, in a paper in Compositio titled "Bounds on canonical Green''s functions", J. Jorgenson and J. Kramer have derived optimal bounds for the hyperbolic and canonical Green''s functions defined on a compact hyperbolic Riemann surface. These estimates were derived in terms of invariants coming from hyperbolic geometry of the Riemann surface. As an application, they deduced bounds for the canonical Green''s functions through covers and for families of modular curves. In this thesis, we extend their methods to noncompact hyperbolic Riemann surfaces and derive similar bounds for the hyperbolic and canonical Green''s functions defined on a noncompact hyperbolic Riemann surface.
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Sumi, Ken. "Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces." Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263432.
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Handfield, Francis Gerald. "Adiabatic limits of the anti-self-dual equation /." Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.
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Taylor, Stephen M. "On Connections Between Univalent Harmonic Functions, Symmetry Groups, and Minimal Surfaces." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1851.pdf.
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Reyes, Ernesto Oscar. "The Riemann zeta function." CSUSB ScholarWorks, 2004. https://scholarworks.lib.csusb.edu/etd-project/2648.
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The Riemann Zeta Function has a deep connection with the distribution of primes. This expository thesis will explain the techniques used in proving the properties of the Rieman Zeta Function, its analytic continuation to the complex plane, and the functional equation that the the Riemann Zeta Function satisfies.
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Boes, Felix Jonathan [Verfasser]. "On moduli spaces of Riemann surfaces : new generators in their unstable homology and the homotopy type of their harmonic compactification / Felix Jonathan Boes." Bonn : Universitäts- und Landesbibliothek Bonn, 2018. http://d-nb.info/1170778070/34.
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Ostermayr, Dominik [Verfasser], Alexander [Gutachter] Alldridge, George [Gutachter] Marinescu, and Tilmann [Gutachter] Wurzbacher. "Some results in supergeometry: Harmonic maps from super Riemann surfaces and Automorphism supergroups of supermanifolds / Dominik Ostermayr ; Gutachter: Alexander Alldridge, George Marinescu, Tilmann Wurzbacher." Köln : Universitäts- und Stadtbibliothek Köln, 2017. http://d-nb.info/1129872475/34.
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Aryasomayajula, Naga Venkata Anilatmaja [Verfasser], Jürg [Akademischer Betreuer] Kramer, Robin de [Akademischer Betreuer] Jong, and Jay [Akademischer Betreuer] Jorgenson. "Bounds for Green's functions on hyperbolic Riemann surfaces of finite volume / Naga Venkata Anilatmaja Aryasomayajula. Gutachter: Jürg Kramer ; Robin de Jong ; Jay Jorgenson." Berlin : Humboldt Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://d-nb.info/1043593225/34.
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Makki, Ali. "Morphismes harmoniques et déformation de surfaces minimales dans des variétés de dimension 4." Thesis, Tours, 2014. http://www.theses.fr/2014TOUR4013/document.
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Dans cette thèse, nous étudions la structure d’un morphisme harmonique F d’une variété riemannienne M4 dans une surface N2 au voisinage d’un point critique mO. Si mO est un point I critique isolé ou si M4 est compact sans bord, nous montrons que F est pseudo-Holomorphe par rapport à une structure presque hermitienne definie dans un voisinage de mO. Si M4 est compact sans bord, les fibres singuliers de F sont des surfaces minimales avec points de branchement. Ensuite, nous étudions des exemples de morphismes harmoniques due a Burel de (S4, gk,l) dans S2 où (gk,I) est une famille de métriques conforme à la métrique canonique. Nous construisons tout d’abord une application semi-Conforme Φk,l de S4 dans S2 en composant deux applications semi-Conformes régulières, F de S4 dans S3 et Φk,i, de S3 dans S2. II résulte de Baird-Eells que le fibres régulier de øk,l pour tout k, I sont minimales. Si [k] = [l] = 1, l’ensemble des points critiques est donnée par l’image réciproque du pâle nord: il consiste en deux 2-Sphères ayant deux points d’intersection. Si k, I 6= 1 l’ensemble des points critiques sont les images réciproques du pôle nord (de la même façon que pour k = t = 1 deux sphères, mais avec une multiplicité I) ainsi que la pré-Image du pôle sud (un tore) avec multiplicité k. Enfin, nous étudions une construction due à Baird-Ou de morphismes harmoniques d’une ensembles ouverts de (S2×S2, can) dans S2. Nous vérifions qu’ils sont holomorphe par rapport à une des quatre structures complexes canoniques hermitiennesIn this thesis, we are interested in harmonic morphisms between Riemannian manifolds (Mm, g) and (Nn, h) for m > n. Such a smooth map is a harmonic morphism if it pulls back local harmonic functions to local harmonic functions: if ƒ : V → ℝ is a harmonic function on an open subset V on N and Φ-1(V) is non-Empty, then the composition ƒ ∘ Φ : Φ-1(V) → ℝ is harmonic. The conformal transformations of the complex plane are harmonic morphisms. In the late 1970's Fuglede and Ishihara published two papers ([Fu]) and ([Is]), where they discuss their results on harmonic morphisms or mappings preserving harmonic functions. They characterize non-Constant harmonic morphisms F : (M,g) → (N,h) between Riemannian manifolds as those harmonic maps, which are horizontally conformal, where F horizontally conformal means : for any x ∈ M with dF(x) ≠ 0, the restriction of dF(x) to the orthogonal complement of kerdF(x) in TxM is conformal and surjective. This means that we are dealing with a special class of harmonic maps
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Javan, Peykar Ariyan. "Explicit polynomial bounds for Arakelov invariants of Belyi curves." Thesis, Paris 11, 2013. http://www.theses.fr/2013PA112075/document.
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On borne explicitement la hauteur de Faltings d'une courbe sur le corps de nombres algèbriques en son degré de Belyi. Des résultats similaires sont démontré pour trois autres invariants arakeloviennes : le discriminant, l'invariant delta et l'auto-intersection de omega. Nos résultats nous permettent de borner explicitement les invariantes arakeloviennes des courbes modulaires, des courbes de Fermat et des courbes de Hurwitz. En plus, comme application, on montre que l'algorithme de Couveignes-Edixhoven-Bruin est polynomial sous l’hypothèse de Riemann pour les fonctions zeta des corps de nombres. Ceci était connu uniquement pour certains sous-groupes de congruence. Finalement, on utilise nos résultats pour démontrer une conjecture de Edixhoven, de Jong et Schepers sur la hauteur de Faltings d'un revêtement ramifié de la droite projective sur l'anneau des entiersWe explicitly bound the Faltings height of a curve over the field of algebraic numbers in terms of the Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound these Arakelov invariants for modular curves, Hurwitz curves and Fermat curves. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithmtime under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we utilize our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a branched cover of the projective line over the ring of integers
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Nualart, Riera Joan. "On the hyperbolic uniformization of Shimura curves with an Atkin-Lehner quotient of genus 0." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/396134.
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The main goal of this thesis is to contribute to the explicit hyperbolic uniformization of Shimura curves. We will restrict to the case of curves attached to Eichler orders in rational quaternion algebras whose maximal Atkin-Lehner quotient has genus 0, which despite multiple differences bears some resemblance to the classical modular case. We will provide an approach to obtain an explicit uniformization of these curves and some of their covers, together with several applications. We will illustrate all the applications with plenty of examples.L’objectiu principal d’aquesta tesi és contribuir a la uniformització hiperbòlica explícita de les corbes de Shimura. Ens restringim a les corbes associades a ordres d’Eichler dins d’àlgebres de quaternions racionals tals que el seu quocient pel grup d’involucions d’Atkin-Lehner és de gènere 0. Aquest cas,tot I que presenta nombroses diferències amb el cas modular clàssic, també hi té certes similituds. Utilitzem aquest fet per a discutir una aproximació al problema de l’obtenció d’uniformitzacions hiperbòliques explícites d’aquestes corbes i d’alguns recobriments, així com també algunes aplicacions, que il·lustrem amb abundants exemples. Per a entendre millor el problema, començarem introduint breument el seu rerefons històric. Després explicarem en detall les nostres contribucions i el contingut de la memòria.
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Shu, Cheng. "E-Polynomial of GLn⋊<σ>-character varieties." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7038.
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Soit σ l'automorphisme par transpose-inverse de GLn, qui définit un produit semi-direct GLn⋊<σ>. Soit Y→X un revê-tement double de surfaces de Riemann, qui est exactement la partie non ramifiée d'un revêtement ramifié de surfaces de Riemann compactes. L'élément non trivial de Gal(Y/X) sera noté τ. A chaque point ramifié enlevé, on associe une GLn(C)-classe de conjugaison contenue dans la composante connexe GLn(C).σ, et on exige que la famille C des classes de conjugaison soient générique. La variété de GLn(C)⋊<σ>-caractère que l'on a étudié est l'espace de module des pairs (L,Φ) formés d'un système local L sur Y et d'un isomorphisme Φ:L → τ*L*, dont les monodromies autour des points ramifiés sont déterminées par C. On calcule le E-polynôme de cette variété de caractère. A ce fin, on utilise un théorème de Katz, ce qui nous ramème au comptage des points sur corps finis. La formule de comptage fait intervenir les caractères irréductibles de GL_n(q)⋊<σ>, et donc la table des l-adic caractères de ce groupe est déterminée au fur et à mesure. Le polynôme qui en résulte s'exprime comme un produit scalaire de certaines fonctions symétriques associées au produit de couronne (Z/2Z)^N⋊(S_N), avec N=[n/2]Let σ be the transpose-inverse automorphism of GLn so that we have a semi-direct product GLn⋊<σ>. Let Y→X be a double covering of Riemann surfaces, which is exactly the unramified part of a ramified covering of compact Riemann surfaces. The non trivial covering transformation is denoted by τ. To each puncture (removed ramification point), we prescribe a GLn(C)-conjugacy class contained in the connected component GLn(C).σ . And we require the collection C of these conjugacy classes to be generic. Our GLn(C)⋊<σ>-character variety is the moduli of the pairs (L,Φ), where L is a local system on Y and Φ:L → τ*L* is an isomorphism, whose monodromy at the punctures are determined by C. We compute the E-polynomial of this character variety. To this end, we use a theorem of Katz and translate the problem to point-counting over finite fields. The counting formula involves the irreducible characters of GL_n(q)⋊<σ>, and so the l-adic character table of GL_n(q)⋊<σ> is determined along the way. The resulting polynomial is expressed as the in-ner product of certain symmetric functions associated to the wreath product (Z/2Z)^N⋊(S_N), with N=[n/2]
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Larsson, David. "Algorithmic Construction of Fundamental Polygons for Certain Fuchsian Groups." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-119916.
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The work of mathematical giants, such as Lobachevsky, Gauss, Riemann, Klein and Poincaré, to name a few, lies at the foundation of the study of the highly structured Riemann surfaces, which allow definition of holomorphic maps, corresponding to analytic maps in the theory of complex analysis. A topological result of Poincaré states that every path-connected Riemann surface can be realised by a construction of identifying congruent points in the complex plane, the Riemann sphere or the hyperbolic plane; just three simply connected surfaces that cover the underlying Riemann surface. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the hyperbolic plane, which is the richest source for Riemann surfaces, these groups are called Fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. What is accomplished in this thesis is a combination of the methods found by Reidemeister & Schreier, Singerman and Voight, and thus provides a unified way of finding Dirichlet domains for subgroups of cofinite groups with a given index. Several examples are considered in-depth.
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Remón, Adell Dionís. "Formes d'ona de Maass i aplicacions = Maass waveforms and applications." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/396139.
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Aquesta memòria està dedicada principalment al tractament computacional de les formes d’ona de Maass i a la consideració d’algunes aplicacions pràctiques derivades del seu estudi. Per abreujar, designarem aquestes funcions, simplement, amb el nom de formes de Maass. Les formes de Maass són funcions infinitament diferenciables que presenten comportaments periòdics (és a dir, automorfs) respecte de grups fuchsians. Des d’un punt de vista numèric, podem dir que les formes de Maass són força més misterioses que les formes automorfes habituals, que són funcions meromorfes. D’aquestes, i especialment quan el grup d’automorfia és un subgrup de congruència del grup modular, se’n coneixen nombrosos exemples numèrics, alguns dels quals es remunten al segle XIX, mentre que ha estat únicament en els darrers anys que s’han obtingut alguns exemples explícits de formes de Maass, referits tots ells a subgrups de congruència del grup modular. D’entrada, la tesi contempla una exposició i una implementació d’algoritmes existents pel càlcul de desenvolupaments a l’entorn de la punta de l’infinit de formes de Maass respecte de subgrups de congruència del grup modular. Tot seguit proposem un conjunt d’algoritmes que, d’acord amb la filosofia de [BT07a] i [BT07b], s’orienten cap a l’obtenció dels desenvolupaments de formes de Maass a l’entorn de punts no necessàriament cuspidals. Aquests algoritmes es tracten en el cas modular i, també, en el cas quaterniònic, en què el grup fuchsià prové de les unitats d’un ordre d’una àlgebra de quaternions racional indefinida. El caràcter discontinu dels grups fuchsians ha estat emprat en el disseny dels anomenats algoritmes de reducció de punts, els quals han resultat bàsics per als objectius anteriors. Al mateix temps, hem fet ús d’aquests algoritmes de reducció de punts pel disseny de codis nous de transmissió de dades en xarxes sense fils i aptes, per tant, per als mòbils que emprem diàriament. Per causa del seu origen, els hem anomenat codis fuchsians.
La memòria està dividida en tres parts i un apèndix. La primera part comprèn del capítol 1 al cap´ıtol 4. Conté una exposició teòrica dels grups fuchsians així com també el desenvolupament d’eines computacionals orientades a les aplicacions posteriors del treball. La segona part comprèn els capítols 5 al 8. En ella presentem les formes de Maass i els conceptes destinats al càlcul dels seus desenvolupaments. La tercera part, que comprèn els capítols 9 i 10, és la dedicada al disseny dels codis fuchsians per a la transmissió de dades. A l’apèndix s’hi troba un resum en anglès.
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Reyes, Edwin Oswaldo Salinas. "Superfícies mínimas de Laguerre e geometria isotrópica." Universidade Federal de Goiás, 2016. http://repositorio.bc.ufg.br/tede/handle/tede/5832.
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In this work we refer to the study of a new method and simple approach to minimal surface Laguerre in isotropic model of Laguerre geometry as the bi-harmonic function graph. We developed the isotropic geometry which studies the geometric properties invariant under certain affine transformations in Euclidean space, and the fundamental elements of Laguerre geometry which are spheres orienteds and plans orienteds, and properties which are invariant on the transformation of Laguerre. In addition, we will show a close relationship between minimal surfaces Laguerre spherical type and isotropic minimal surfaces which are given by the graph of harmonic functions and minimal Euclidean surfaces. Finally, the duality metric in the isotropic space is used to develop an isotropic exchange for minimal surfaces Laguerre in certain Lie transformation of Laguerre minimal surfaces in Euclidean space.
Neste trabalho nos referimos ao estudo de um novo método de desenvolvimento de superfícies mínimas de Laguerre vista no modelo isotrópico da geometria de Laguerre como o gráfico de funções bi-harmônicas. Desenvolvemos a geometria isotrópica a qual estuda as propriedades geométricas invariantes por certas transformações afines no espaço Euclidiano, os elementos fundamentais da geometria de Laguerre as quais são esferas e planos orientados e as propriedades as quais são invariantes sobre as transformações de Laguerre. Além disso, mostraremos uma relação fechada entre superfícies mínimas de Laguerre do tipo esférico e superfícies mínimas isotrópicas as quais são dadas pelo gráfico de funções harmônicas e superfícies mínimas Euclidianas. Finalmente, a métrica dual no espaço isotrópico é utilizada para desenvolver uma contrapartida isotrópica de superfícies mínimas de Laguerre em certas transformações de Lie de superfícies mínimas de Laguerre no espaço Euclidiano.
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"Harmonic maps on surfaces." 1999. http://library.cuhk.edu.hk/record=b5890049.
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by Tsui Wai-kwok Ricky.Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.
Includes bibliographical references (leaves 58-59).
Abstracts in English and Chinese.
Chapter 1 --- Preliminary --- p.4
Chapter 1.1 --- Introduction --- p.4
Chapter 1.2 --- Some basic theorem --- p.7
Chapter 2 --- Bubble tree Convergence for a sequence of harmonic map --- p.11
Chapter 3 --- Heat Flow of Harmonic Maps on Riemann Surface --- p.21
Chapter 3.1 --- Existence of unique solution to the evolution problem --- p.21
Chapter 3.1.1 --- Some Basic Estimates --- p.22
Chapter 3.1.2 --- Existence Result --- p.34
Chapter 3.1.3 --- Behaviour of solutions near singular points --- p.37
Chapter 3.2 --- Finite time Blow-up --- p.39
Chapter 3.3 --- Energy Identity --- p.51
Bibliography --- p.58
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Philip, Eliza. "Function Theory On Non-Compact Riemann Surfaces." Thesis, 2012. http://etd.iisc.ernet.in/handle/2005/2330.
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The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. The theory splits in two parts; the compact and the non-compact case. The function theory developed on these cases are quite dissimilar. The main difficulty one encounters in the compact case is the scarcity of global holomorphic functions, which limits one’s study to meromorphic functions. This however is not an issue in non-compact Riemann surfaces, where one enjoys a vast variety of global holomorphic functions. While the function theory of compact Riemann surfaces is centered around the Riemann-Roch theorem, which essentially tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles, the function theory developed on non-compact Riemann surface engages tools for approximation of functions on certain subsets by holomorphic maps on larger domains. The most powerful tool in this regard is the Runge’s approximation theorem. An intriguing application of this is the Gunning-Narasimhan theorem, which says that every connected open Riemann surface has an immersion into the complex plane. The main goal of this project is to prove Runge’s approximation theorem and illustrate its effectiveness in proving the Gunning-Narasimhan theorem. Finally we look at an analogue of Gunning-Narasimhan theorem in the case of a compact Riemann surface.
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Tai, Tien-En, and 戴天恩. "The Theory of Riemann Surfaces and Elliptic Functions with application to Nonlinear Schrodinger Equation." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/66233520264950696666.
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碩士國立清華大學
數學系
101
In this paper, we study the theory of Riemann Surfaces of genus N and the numerical computation of path integrals on those Riemann Surfaces. We then study the classical theory of the elliptic functions. Finally, we apply both theorys to solve some special solutions of Nonlinear Schrodinger Equation.
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Tzeng, Tz-Shiuan, and 曾子軒. "The Theory of Riemann Surfaces and Elliptic Functions with application to the sine-Gordon Equations." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/08924820148317943844.
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碩士國立清華大學
數學系
101
In this paper, we study the theory of Riemann Surfaces of ge-nus N and the numerical computation of path intergrals on those Riemann Surfaces. We then study the classical theory of the Eillptic functions. Finally, we apply both theorys to solve some special solutions of the sine-Gordon Equation.
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Jiang, Yi-Chin, and 蔣宜津. "The Theory of Riemann Surfaces and Elliptic Functions with Application to the Nonlinear Schrodinger Equation." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/86358559767396519384.
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碩士國立交通大學
應用數學系所
101
In this paper, we study the function theory of the solutions of the nonlinear Schrodinger equation (NLS), and these solutions have the radical forms. Solutions of such equation resides on the the Riemann surface of genus N-1 so we first study the theory of Riemann surface. Then we study the classical elliptic functions to solve some special solutions of NLS and analyze the associated properties.
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Chen, Jian-Ze, and 陳建澤. "The Theories of Riemann Surfaces and Elliptic Functions with Application to the sine-Gordon Equation." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/13014433361767114790.
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碩士國立交通大學
應用數學系所
101
The Goal of this paper is to solve the sine-Gordon equation, u_tt - u_xx + sin[u(x,t)] = 0, where -∞ < x < ∞ and t > 0. By using the method of substitution, we get u_ss + sin[u(s)] = 0, which is a simple pendulum motion at time s with the angular displacement u, and it implies u_s = √2[E + cos(u)], where E is constant. But √2[E + cos(u)] is a two-valued function on C, so we introduce the theory of the Riemann surface R such that it comes to a single-valued analytic function on this surface. Next, we introduce the classical theory of the elliptic functions, to solve u_ss + sin[u(s)] = 0, and analyze the associated properties.
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Rowland, Todd. "Smooth holomorphic curves in S [superscript 6] /." 1999. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:9943108.
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Huang, Jian-Shun, and 黃建順. "The Theory of Riemann Surfaces and the Weierstrass Elliptic Functions with Application to the Korteweg-deVries Equation." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/78897276527866821382.
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碩士國立交通大學
應用數學系所
104
The Korteweg-deVries equation is nonlinear partial differential equations, and the KdV equation is as follows: u_t(x,t)-6*u(x,t)u_x(x,t)+u_xxx(x,t)=0,t>0,-∞≤x≤∞ For traveling solutions, we can transform partial differential equations into differential equation, and the KdV equation becomes the following form: u_θ^2(θ)=2*u^3(θ)+cu^2(θ)+2*A*u(θ)+B To solve u(θ) we transfer this ode into integral equation namely, The inverse problem where the integral involves square root(a multi-valued function). We develop Riemann surfaces with proper algebraic structure to make the function √ to be single-valued. Then we introduce the classical theory of Weierstrassian elliptic functions, to solve the solution of u_θ(θ)=√(2*u^3(θ)+cu^2(θ)+2*A*u(θ)+B) .
26
Ram, Mohan Devang S. "An Introduction to Minimal Surfaces." Thesis, 2014. http://etd.iisc.ernet.in/handle/2005/2890.
Full textAbstract:
In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform a given map to a harmonic map (i.e., find a harmonic map in its homotopy class). We follow J¨urgen Jost’s approach using classical potential theory techniques. Subsequently, we analyze the additional conditions needed to ensure a certain uniqueness property of harmonic maps within a given homotopy class. In conclusion, we look at a couple of applications of what we have shown thus far and we find a neat proof of a slightly weaker version of Hurwitz’s Automorphism Theorem.
In the second chapter, we introduce the concept of minimal surfaces. After exploring a few examples, we mathematically formulate Plateau’s problem regarding the existence of a soap film spanning each closed, simple wire frame and discuss a solution. In conclusion, a partial result (due to Rad´o) regarding the uniqueness of such a soap film is discussed.