Now we’re going to use the machinery already developed to prove the existence of harmonic functions.

Fix a Riemann surface {M} and a coordinate neighborhood {(U,z)} isomorphic to the unit disk {D_1} in {\mathbb{C}} (in fact, I will abuse notation and identify the two for simplicity), with {P \in M} corresponding to {0}.

First, one starts with a function {h: D_1 - \{0 \} \rightarrow \mathbb{C}} such that:

1. {h} is the restriction of a harmonic function on some {D_{1+\epsilon} - 0} 2. {d {}^* h = 0} on the boundary {\partial D_1} (this is a slight abuse of notation, but ok in view of 1).

The basic example is {z^{-n} + \bar{z}^{n}}.

Theorem 1 There is a harmonic function {f: M - P \rightarrow \mathbb{C}} such that {f-h} is continuous at {P}, and {\phi df \in L^2(M)} if {\phi} is a bounded smooth function that vanishes in a neighborhood of {P}.

 

In other words, we are going to get harmonic functions that are not globally defined, but whose singularities are localized.

The differential {\theta}

 

Define the function {\tilde{h} : M \rightarrow \mathbb{C}} such that {\tilde{h} = h} in {U -D_{1/2}}, {\tilde{h}} is smooth in {U}, and {\tilde{h}=0} outside {U}. Define

\displaystyle \theta(x) = d \tilde{h}(x) := \begin{cases} d \tilde{h} \ \mathrm{if} \ x \in U \\ 0 \ \mathrm{otherwise} \end{cases}

Evidently {\theta \in L^2(M)}, so we can write

\displaystyle \theta = \alpha + \beta + \omega

where {\alpha \in E, \beta \in E^*, \omega \in H}.

Now, by yesterday’s second result, we find that {\alpha} is smooth in {U}. We are going to study {\alpha} further.

Harmonicity of {\alpha} outside {D_{1/2}}

 

I claim, moreover, that {\alpha} is in fact harmonic outside of {D_{1/2}}. To see this, choose {\phi} a smooth real-valued function supported in {M-D_{1/2}}, and consider

\displaystyle ( d \phi, \alpha)_M = ( d \phi, \theta)_M = ( d \phi, \theta)_{D_1 - D_{1/2}}

by the way {\theta} was defined. This is

\displaystyle I = \int_{D_1 - D_{1/2}} d\phi \wedge {}^* \bar{\theta},

which I claim vanishes. To see this, consider the 1-form {\omega' = \phi {}^* \bar{\theta}}; then

\displaystyle d \omega' = d \phi \wedge {}^* \bar{\theta} - \phi d {}^* \theta,

so by Stokes’s theorem

\displaystyle I = \int_{\partial(D - D_{1/2})} \omega' + \int \int \phi d {}^* \theta.

By harmonicity of {h} the second term vanishes. The first term does too—the integral over {C_{1/2}} vanishes because of the {\phi} in {\omega'}; on {C_1} we have {{}^* \theta = 0} by assumption.

Conclusion of the proof

 

So {\alpha} is everywhere smooth, thus by the previous post, exact. We can write {\alpha = dg}; then consider

\displaystyle f := g - \tilde{h} + h

which is extended to all of {M-\{P\}} by equalling {g} outside of {D_1} (so it is continuous). Then {f} equals {g} outside of {D_{1/2}}, in fact, so is harmonic there. Inside {D_{1/2} - \{P\}}:

\displaystyle df = \alpha - \theta + dh = - \beta - \omega - dh

which is necessarily smooth and also orthogonal to closed forms supported in {D_{1/2}}, so co-closed there. Thus {f} is harmonic everywhere. This is the desired function.

Applications

 

These are the basic existence results for harmonic/meromorphic functions.

Theorem 2 Given {P} in {M}, {n \geq 1}, there is a harmonic function {u: M - \{P\} \rightarrow \mathbb{C}}, with {u - Re(z^{-n})} continuous in a neighborhood of {P} if {z} is a fixed local coordinate.

Apply the previous result with {h = z^{-n} + \bar{z}^n}.

A meromorphic differential associates to each coordinate system {(z,U)} a meromorphic function {f} such that {f(z) dz} is invariant under changes of coordinates in the usual way.

Theorem 3 Hypotheses as above, there is a meromorphic differential {\omega} on {M} which is holomorphic on {M- \{P\}} and in a neighborhood of {P} is\displaystyle \frac{dz}{z^n }

if {n >1}. 

Take {\omega = du + i {}^* du} if {u} is as above for {n-1} replacing {n}.

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.