Complex manifolds of dimension 1

Page of the course:
http://verbit.ru/IMPA/RS-2024/

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.

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. Examples of Riemann surfaces: Riemann surface
of a function, plane curves, complex tori.

4. Hyperbolic geometry, Poincare metric, geodesics,
Moebius group and its action. Discrete subgroups of the Moebius
group: Fuchsian group, Kleinian group and their geometry.

5. Normal families. Montel's theorem. Arzela-Ascoli theorem.
Riemann uniformization theorem.

6. Schwarz lemma. Group of automorphisms of a disk.
Poincare metric. Kobayashi metric. Existence of constant
curvature metrics on Riemann surfaces.

7. Elliptic equations. Strong maximum principle.
Local solutions of elliptic equations on a domain with
boundary (existence and uniqueness).

8. Formal integrability of almost complex structures.
Nijenhuis tensor. Integrability in complex dimension 1.

9 (*). Riemann-Roch formula in dimension 1.
Uniqueness of the complex structure on a 2-sphere,
with applications to the diffeomorphism group.

10 (*). Poincare-Koebe uniformization theorem.

11 (*). Quasi-conformal maps and the Teichm\"uller space.

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 (*)
will be offered only if time permits.