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Decomposition of the space of square-integrable 1-forms

Posted by Akhil Mathew under

analysis,

complex analysis | Tags:

1-forms,

harmonic forms,

Hodge theory,

Riemann surfaces,

Weyl's lemma |

1 Comment
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.

**Computation of **

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. *

**Computation of **

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

Also

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.

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