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 1There 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 2Given 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 3Hypotheses as above, there is a meromorphic differential on which is holomorphic on and in a neighborhood of isif .

Take if is as above for replacing .

Theorem 4There 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.