On Connections Between Univalent Harmonic Functions, Symmetry Groups, and Minimal Surfaces
We survey standard topics in elementary differential geometry and complex analysis to build up the necessary theory for studying applications of univalent harmonic function theory to minimal surfaces. We then proceed to consider convex combination harmonic mappings of the form f=sf_1+(1-s) f_2 and g...
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| Format: | Others |
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BYU ScholarsArchive
2007
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| Online Access: | https://scholarsarchive.byu.edu/etd/1003 https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=2002&context=etd |
| Summary: | We survey standard topics in elementary differential geometry and complex analysis to build up the necessary theory for studying applications of univalent harmonic function theory to minimal surfaces. We then proceed to consider convex combination harmonic mappings of the form f=sf_1+(1-s) f_2 and give conditions on when f lifts to a one-parameter family of minimal surfaces via the Weierstrauss-Enneper representation formula. Finally, we demand two minimal surfaces M and M' be locally isometric, formulate a system of partial differential equations modeling this constraint, and calculate their symmetry group. The group elements generate transformations that when applied to a prescribed harmonic mapping, lift to locally isometric minimal surfaces with varying graphs embedded in mathbb{R}^3. |
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