# Mr Kyle Talbot**Honours graduate**
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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.