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.
Define the function such that in , is smooth in , and outside . Define
Evidently , so we can write
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.
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
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.