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.