I realize I’ve been slow as of late.
Though the decomposition of the square-integrable 1-forms was a Hilbert space, -decomposition, it reflects many smoothness properties in which we are interested. The goal of this post is twofold: first, show that the smooth differentials in the first two components are respectively the exact and coexact ones; and second, show that the closedness of a square-integrable 1-form in some neighborhood implies the smoothness of the corresponding term. The ultimate goal is the existence theorems for harmonic functions on Riemann surfaces.
Smooth differentials in
I now claim that any smooth differential is exact and any smooth differential in is co-exact. This is nontrivial because of the way we took completions. I will only prove the claim for .
We already have closedness of by the previous post. To show exactness, it will be sufficient to show that for any closed smooth curve ,
because we could use a path integral to define the antiderivative. We can approximate by a simple closed curve homotopic to , so we can assume at the outset that is a homeomorphism .
Proposition 1 Given as above, there is a closed differential such that
for any closed .
To prove this, find an annular neighborhood of the compact set , which is divided into as in the figure.
Choose smooth on taking the value in a neighborhood of the outside part of , on a neighborhood of the inside part of .Define the differential
by Stokes theorem and since is closed. But this is
proving the proposition.
Now fix a smooth ; from its orthogonality to and smoothness is seen to be exact.
Smoothness of components
Fix a differential , say
where . We want conditions that will give local smoothness of in case is in some neighborhood; is always smooth.
Proposition 2 If is smooth and closed in a neighborhood , so is .
This is local, so we can assume is a coordinate neighborhood with coordinate . Write
We are given that are smooth; we need to prove that are. Take any smooth supported in and consider
The first equals . The second is zero because is closed in . So
But we also have
if we apply the same reasoning to itself! So as a result is harmonic in , hence smooth, and so is . The other part of the proof is similar (replace with , etc.).