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.
| Identifer | oai:union.ndltd.org:IISc/oai:etd.ncsi.iisc.ernet.in:2005/2890 |
| Date | January 2014 |
| Creators | Ram Mohan, Devang S |
| Contributors | Verma, Kaushal |
| Source Sets | India Institute of Science |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
| Relation | G26306 |
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