Mr Patrick Korbel
Doctor of Philosophy student
Honours graduate
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Office: 744 | Telephone: +61 8 8313 1606
Research seminars
Doctoral thesis
Mathematics education in Australia
Honours thesis
The Uniformisation Theorem
The aim of this thesis is to classify all simply connected and connected two-dimensional
manifolds with conformal metric (Riemann surfaces) via a thorough and annotated proof
of the Uniformisation Theorem. The statement of the theorem is that every simply
connected and connected Riemann surface is biholomorphic (holomorphic with inverse)
either to the Riemann sphere (the union of the complex plane and the point at infinity),
the complex plane or the unit disk in the complex plane. The proof is based on a proof
of the same theorem given by Jean-Pierre Demailly (see Appendix A).
Firstly the Riemann Mapping Theorem will be proven after finding the automorphisms
of the Riemann sphere, complex plane and unit disk, and proving Montelâs Theorem.
The Riemann Mapping Theorem shows that a connected and simply connected open
subset of the complex plane is either equal to the complex plane itself or biholomorphic
to the unit disk.
Then it is shown that all compact, simply connected Riemann surfaces are biholomor-
phic to the Riemann sphere. This is done by solving a certain differential equation using
functional analysis techniques on L p spaces. Finally, the proof of the Uniformisa-
tion Theorem is completed by showing that a non-compact simply connected Riemann
surface can be embedded as an open subset of the complex plane.
