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1	An Introduction to Minimal Surfaces Ram Mohan, Devang S January 2014 (has links) (PDF) 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. 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