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An Introduction to Minimal Surfaces

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.

Identiferoai:union.ndltd.org:IISc/oai:etd.ncsi.iisc.ernet.in:2005/2890
Date January 2014
CreatorsRam Mohan, Devang S
ContributorsVerma, Kaushal
Source SetsIndia Institute of Science
Languageen_US
Detected LanguageEnglish
TypeThesis
RelationG26306

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