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Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition

$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\mathrm{d})\oplus\mathrm{ran}(\delta)$$

where $\mathcal{H}^{k}(\mathcal{M})$ denotes the space of harmonic $k$-forms.

Are there any known results for compactly-supported forms in the non-compact case?

For the non-compact case, the only relevant theorem I could find is due to Kodaira in the case $(\mathcal{M},g)$ is complete and reads

$$\Omega^{k}_{L^{2}}(\mathcal{M})\cong\mathcal{H}^{k}_{L^{2}}(\mathcal{M})\oplus\overline{\mathrm{ran}(\mathrm{d}\vert_{\Omega^{k-1}_{c}})}^{\Vert\cdot\Vert_{L^{2}}}\oplus\overline{\mathrm{ran}(\delta_{\Omega^{k+1}_{c}})}^{\Vert\cdot\Vert_{L^{2}}}$$

where $\Omega^{k}_{L^{2}}(\mathcal{M})$ is the space of smooth $k$-forms, which are also in $L^{2}$. See also reference (2) below. However, in my case, I am only interested in the case of $\Omega^{k}_{c}(\mathcal{M})$, which is a subspace of $\Omega^{k}_{L^{2}}(\mathcal{M})$. Furthermore, can one relax the completeness assumption in this case?

References:

  1. K. Kodaira: Harmonic fields in riemannian manifolds (generalized potential theory). Annals of Mathematics 50, pages 587–665, 1949.
  2. E. Amar: On the $L^{r}$ Hodge theory in complete non compact Riemannian manifolds, Mathematische Zeitschrift (287), pages 751–795, 2017.
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    $\begingroup$ How would this work for $\mathbb{R}$ with the standard metric? Presumably, $\mathcal{H}^1$ should be translation invariant, but there is no translation invariant compactly supported $1$-form. (Recall that $H^1_c$ of the line is $1$-dimensional.) $\endgroup$
    – David E Speyer
    Mar 21, 2023 at 12:25
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    $\begingroup$ @DavidESpeyer: Harmonic contribution would be zero, just like in the $L^2$ case. $\endgroup$
    – Moishe Kohan
    Mar 21, 2023 at 14:15
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    $\begingroup$ Okay, so the goal is not to give representatives for $H^k_c$ ? $\endgroup$
    – David E Speyer
    Mar 21, 2023 at 14:29
  • $\begingroup$ Exactly. The goal is to see whether it is possible to decompose a compactly supported k-form in a similar way as on compact manifolds. $\endgroup$
    – B.Hueber
    Mar 21, 2023 at 17:25
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    $\begingroup$ The issue is that for $\mathbb R$ with the standard metric the image of $d$ is a cxdimension $1$ subspace, so the only way to get a decomposition of this form is to sum it with a $1$-dimensional subspace, but there is no canonical (i.e. translation-invariant) $1$-dimensional subspace, so no decomposition of this form can exist without replacing $d$ by something very different. $\endgroup$
    – Will Sawin
    Mar 21, 2023 at 21:06

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