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,
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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.
December 12, 2009 at 10:45 pm
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