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August 2019
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Mr Kyle Talbot

Honours graduate

 

Office: 721 |


Honours thesis

Constructing constant scalar curvature metrics on surfaces

This thesis considers the problem of constructing constant scalar curvature metrics on surfaces by conformal changes in the metric, which is motivated by a geometric classifica- tion of surfaces via the well-known uniformization theorem. The problem is shown to be reducible to the existence of a solution to a nonlinear partial differential equation defined over a given surface satisfying suitable hypotheses. Using the theory of linear and nonlin- ear elliptic differential operators developed throughout, the equation is analysed directly for zero- and negative scalar curvature and existence proofs are given using standard lin- ear theory and the continuity method respectively. Positive scalar curvature is addressed briefly in light of recent work by Chen, Lu and Tian using the Ricci flow.