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. (more…)