Harmonic Maps of Finite type vs. Finite Uniton Number

Who: Simon Bonnerup Pedersen
When: Wednesday, December 19, 2012 at 16:15
Where: U49

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Simon Bonnerup Pedersen defends his Master Thesis on Harmonic Maps of Finite type vs. Finite Uniton NumberSimons Master thesis advisor was Martin Svensson.


In this master thesis we study harmonic maps and investigate how they arise from certain holomorphic maps into loop groups via the DPW method using the concept of Riemannian symmetric spaces. The approach to this method uses a lot of algebraic group theory which turns out to be much simpler than solving the corresponding harmonic differential equations analytically. We describe how the process of adding a uniton in these loop groups works, and we will observe that this leads us to a method where we can construct new harmonic maps from old ones. By introducing harmonic maps of finite uniton number we will be able to classify harmonic maps from Riemann surfaces into certain symmetric spaces. We shall in particularly be interested in developing a procedure to construct harmonic maps into a Grassmannian. We introduce another class of harmonic maps, namely harmonic maps of finite type arising from a certain class of constant holomorphic potentials in the DPW approach. Following Pacheco as in [14] we finally study harmonic maps which are simultaneously of finite type and of finite uniton number, and show that such harmonic maps from a torus into CPn are constant maps. Furthermore we discuss some generalisations of this result.

The thesis is available for download in PDF format (in Danish).