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Function Theory On Non-Compact Riemann Surfaces

Electronic Theses of Indian Institute of Science

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Title Function Theory On Non-Compact Riemann Surfaces
 
Creator Philip, Eliza
 
Subject Complex Functions
Non-compact Reimann Surfaces
Runge's Approximation Theorem
Cohomology
Riemann Surfaces
Differential Forms
Sheaf Theory
Geometry
 
Description 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.
 
Contributor Verma, Kaushal
 
Date 2014-06-30T05:19:45Z
2014-06-30T05:19:45Z
2014-06-30
2012-05
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2330
http://etd.ncsi.iisc.ernet.in/abstracts/2996/G25292-Abs.pdf
 
Language en_US
 
Relation G25292