The theorems of Picard: The Poincare metric on the unit disk. Conformal metrics on domains and
curvature. The Ahlfors--Schwarz lemma. Existence of a conformal metric on C\{0,1} with negative
curvature. Normal families and Montel’s theorem. Picard’s little and big theorems.
The Riemann mapping theorem: Harmonic and subharmonic functions. Equivalent characterisations of
subharmonic functions. Harnack’s inequality. The Perron method for solving the Dirichlet problem.
Barriers and sufficient condition for solvability. Green’s functions. Existence of Green’s function. The
Riemann mapping theorem using Green’s functions. Equivalent characterisations of simply-connected
domains.
Analytic continuations and Monodromy theorem: Analytic continuation along curves. Permanence of
relations. The sheaf of germs of a holomorphic function. The complete analytic function of a germ. The
monodromy theorem. Relation to covering spaces. The Riemann surface of an algebraic function.