Yesterday I defined the Hilbert space of square-integrable 1-forms on a Riemann surface . Today I will discuss the decomposition of it. Here are the three components:
1) is the closure of 1-forms where is a smooth function with compact support.
2) is the closure of 1-forms where is a smooth function with compact support.
3) is the space of square-integrable harmonic forms.
Today’s goal is:
Theorem 1 As Hilbert spaces,
The proof will be divided into several steps.
I claim that the smooth, compactly supported forms orthogonal to are precisely the co-closed ones. Indeed, if is smooth and compactly supported, and is arbitrary:
By the identities proved yesterday (essentially Stokes formula) this is
It is now clear that this vanishes for all smooth and compactly supported if f .
Similarly the smooth 1-forms orthogonal to are precisely the closed ones.
Corollary 2 are orthogonal.
This is the key place where we will use Weyl’s lemma. Let be an arbitrary coordinate neighborhood with coordinate . Let , and suppose on .
Lemma 3 is smooth in .
As a result, we see by the previous section that is closed and co-closed, so harmonic.
Now we prove the lemma.
If is an arbitrary smooth real function supported in , then
Both these equal zero though by assumption. Subtracting yields
which since was arbitrary implies harmonicity of by Weyl’s lemma. A similar argument with shows the same thing for . As a result, we get smoothness.