MATH-F-513: Riemann surfaces
Riemann surface is a complex manifold of dimension 1.
Equivalently, Riemann surface is a 2-dimensional smooth
manifold with a fixed conformal structure. The "conformal
structure" is a Riemannian structure, defined
up to a multiplication by a positive function,
that is, a way of defining "angles" between
tangent vectors. Equivalence of these two definitions
is a non-trivial theorem, and one of the main
results of the course. Most of the time, we will
discuss complex geometry, with a special emphasis
on dimension 1, but there is a bunch of results
which are special for Riemannian surfaces alone.
These are "uniformization theorems", which can be
understood as existence of canonical (and very
symmetric) complete metric on any Riemann surface.
Using this metric, all Riemann surfaces except
three are obtained as quotients of the Poincare
plane by a discrete group of isometries. I will
try to explain the relevant complex geometry and some
of the uniformization theorems.
Program
1. Almost complex structures and Hodge decomposition.
Cauchy-Riemann equation via the Hodge decomposition.
Integrability of almost complex structures.
Existence of holomorphic function as a sufficient
condition for integrability (dimension 1).
2. Complex manifolds, sheaves of holomorphic
functions, equivalence of definitions of a complex manifold
(definition via integrable almost complex structure,
definition using sheaves, definition through charts
and atlases).
3. Real analytic manifolds. Formal integrability of
almost complex structures. Newlander-Nirenberg theorem
for real analytic manifolds.
4. Examples of Riemann surfaces: Riemann surface
of a function, plane curves, complex tori.
5.(*) Weierstrass p-function. Elliptic curve and its equation.
j-invariant of an elliptic curve.
6. Normal families. Montel's theorem. Arzela-Ascoli theorem.
Riemann uniformization theorem.
7. Schwartz lemma. Group of automorphisms of a disk.
Poincare metric. Kobayashi metric. Existence of constant
curvature metrics on Riemann surfaces.
8.(*) Iterated holomorphic mappings. Fatou and Julia:
dynamics on Riemann sphere. Fractal sets. Mandelbrot sets.
Knowledge of geometry (manifolds, tangent bundles,
Riemann structures) and complex analysis (Cauchy formula,
analytic continuation, complex differentiable and complex
analytic functions) is required. Parts marked with (*)
would be possibly omitted for the lack of time.
Literature:
John W Milnor, "Dynamics in one complex variable",
Annals of mathematics studies, no. 160,
Princeton University Press
Lars V. Ahlfors, "Lectures on Quasiconformal Mappings",
University Lecture Series 38, American Mathematical
Society, 2006.
Simon Donaldson, "Riemann Surfaces",
Oxford Graduate Texts in Mathematics 22,
Oxford University Press, 2011.
C Herbert Clemens, "A scrapbook of complex curve theory"
University series in mathematics (Plenum Press)
Plenum Press, New York, 1980.