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An Introduction to Minimal Surfaces

Electronic Theses of Indian Institute of Science

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Title An Introduction to Minimal Surfaces
 
Creator Ram Mohan, Devang S
 
Subject Minimal Surfaces
Riemann Surfaces
Harmonic Maps
Plateau's Problem
Riemannian Metric
Hilbert Space
Sobolev Space
Energy of a Map
Weingarten Map
Catenoid
Helicoid
Enneper Surface
Hurwitz's Automorphism Theorem
Geometry
 
Description 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.
 
Contributor Verma, Kaushal
 
Date 2017-12-10T08:22:15Z
2017-12-10T08:22:15Z
2017-12-10
2014
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2890
http://etd.ncsi.iisc.ernet.in/abstracts/3752/G26306-Abs.pdf
 
Language en_US
 
Relation G26306