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…)