Now we’re going to use the machinery already developed to prove the existence of harmonic functions.
Fix a Riemann surface and a coordinate neighborhood
isomorphic to the unit disk
in
(in fact, I will abuse notation and identify the two for simplicity), with
corresponding to
.
First, one starts with a function such that:
1. is the restriction of a harmonic function on some
2.
on the boundary
(this is a slight abuse of notation, but ok in view of 1).
The basic example is .
Theorem 1 There is a harmonic function
such that
is continuous at
, and
if
is a bounded smooth function that vanishes in a neighborhood of
.
In other words, we are going to get harmonic functions that are not globally defined, but whose singularities are localized.
The differential
Define the function such that
in
,
is smooth in
, and
outside
. Define
Evidently , so we can write
where .
Now, by yesterday’s second result, we find that is smooth in
. We are going to study
further.
Harmonicity of outside
I claim, moreover, that is in fact harmonic outside of
. To see this, choose
a smooth real-valued function supported in
, and consider
by the way was defined. This is
which I claim vanishes. To see this, consider the 1-form ; then
so by Stokes’s theorem
By harmonicity of the second term vanishes. The first term does too—the integral over
vanishes because of the
in
; on
we have
by assumption.
Conclusion of the proof
So is everywhere smooth, thus by the previous post, exact. We can write
; then consider
which is extended to all of by equalling
outside of
(so it is continuous). Then
equals
outside of
, in fact, so is harmonic there. Inside
:
which is necessarily smooth and also orthogonal to closed forms supported in , so co-closed there. Thus
is harmonic everywhere. This is the desired function.
Applications
These are the basic existence results for harmonic/meromorphic functions.
Theorem 2 Given
in
,
, there is a harmonic function
, with
continuous in a neighborhood of
if
is a fixed local coordinate.
Apply the previous result with .
A meromorphic differential associates to each coordinate system a meromorphic function
such that
is invariant under changes of coordinates in the usual way.
Theorem 3 Hypotheses as above, there is a meromorphic differential
on
which is holomorphic on
and in a neighborhood of
is
if
.
Take if
is as above for
replacing
.
Theorem 4 There are non-constant meromorphic functions on any Riemann surface.
Take the ratio of any two meromorphic differentials as in the previous theorem with poles at different points.
Ok, where next? I admit I’m not a huge fan of this approach, and I’ll probably use a sheafier method when I start (next) talking about the buildup to the Riemann-Roch theorem.
December 14, 2009 at 7:13 pm
Hi!
Great series of posts. What will you discuss using sheaf theory?
December 14, 2009 at 7:22 pm
So far I wasn’t aiming for too much. I’ll probably outline the basic facts about sheaf cohomology and prove the de Rham/Dolbeaut theorems. The main goal was to get a clean framework for doing the part of the Riemann-Roch theorem that can be done without Serre duality, though this really can be done without sheaves.
Of course, in the future I will probably refer back heavily to the sheaf material for complex analytic geometry and regular algebraic geometry as well.