Proof of the Hodge Decomposition via Elliptic Theory

The Hodge decomposition rests on the theory of elliptic operators on compact manifolds. We present the key analytical ingredients and show how they combine to yield the decomposition.

Elliptic Operators on Compact Manifolds

Definition10.7Elliptic operator

A linear differential operator $P: \Gamma(E) \to \Gamma(F)$ of order $m$ between vector bundles is elliptic if its principal symbol $\sigma_m(P)(x, \xi): E_x \to F_x$ is an isomorphism for all $x \in M$ and $\xi \in T^*_xM \setminus \{0\}$ .

The Hodge Laplacian $\Delta = d\delta + \delta d$ has principal symbol $\sigma_2(\Delta)(x, \xi) = |\xi|^2 \mathrm{Id}$ , which is positive definite for $\xi \neq 0$ . Hence $\Delta$ is elliptic.

Key Analytical Results

Theorem10.9Elliptic regularity

If $P$ is an elliptic operator of order $m$ on a compact manifold and $Pu = f$ with $f \in H^s$ (Sobolev space), then $u \in H^{s+m}$ . In particular, if $f \in C^\infty$ , then $u \in C^\infty$ . Applied to $\Delta$ : any distributional solution of $\Delta\omega = f$ with $f$ smooth is itself smooth.

Theorem10.10Fredholm property

An elliptic operator $P: H^{s+m}(E) \to H^s(F)$ on a compact manifold is Fredholm: it has finite-dimensional kernel, closed range, and finite-dimensional cokernel. The index $\mathrm{ind}(P) = \dim\ker P - \dim\mathrm{coker}\, P$ is a topological invariant.

The Hodge Decomposition Proof

Proof

We prove the decomposition $\Omega^k = \mathcal{H}^k \oplus d\Omega^{k-1} \oplus \delta\Omega^{k+1}$ in detail.

Step 1: The orthogonal complement of $\ker\Delta$ . Since $\Delta$ is self-adjoint and elliptic on a compact manifold:

$\ker\Delta$ is finite-dimensional (by the Fredholm property).
$L^2\Omega^k = \ker\Delta \oplus \overline{\mathrm{im}(\Delta)}$ (since $\Delta$ is self-adjoint, $(\ker\Delta)^\perp = \overline{\mathrm{im}(\Delta)}$ ).
The image is closed (Fredholm property), so $\overline{\mathrm{im}(\Delta)} = \mathrm{im}(\Delta)$ .

Thus $L^2\Omega^k = \ker\Delta \oplus \mathrm{im}(\Delta)$ , and by elliptic regularity (smooth inputs give smooth outputs), this holds at the smooth level: $\Omega^k = \mathcal{H}^k \oplus \Delta\Omega^k$ .

Step 2: Splitting $\mathrm{im}(\Delta)$ . We claim $\Delta\Omega^k = d\Omega^{k-1} \oplus \delta\Omega^{k+1}$ (orthogonal).

Orthogonality: $\langle d\alpha, \delta\beta \rangle = \langle d^2\alpha, \beta \rangle = 0$ since $d^2 = 0$ .

Inclusion $\supset$ : $d\alpha = d(\delta d + d\delta)G\alpha'$ (using the Green's operator), and similarly for $\delta$ . In fact, for $\omega \perp \mathcal{H}^k$ , we have $\omega = \Delta G\omega = d\delta G\omega + \delta d G\omega$ , where $d\delta G\omega \in d\Omega^{k-1}$ and $\delta d G\omega \in \delta\Omega^{k+1}$ .

Inclusion $\subset$ : $\Delta\omega = (d\delta + \delta d)\omega = d(\delta\omega) + \delta(d\omega) \in d\Omega^{k-1} + \delta\Omega^{k+1}$ .

Step 3: Assembling the decomposition. From Steps 1 and 2:

\Omega^k = \mathcal{H}^k \oplus d\Omega^{k-1} \oplus \delta\Omega^{k+1},

with pairwise orthogonality. Every $\omega \in \Omega^k$ decomposes uniquely as $\omega = H\omega + d\alpha + \delta\beta$ , where $H$ is the harmonic projection, $\alpha = \delta G\omega$ , and $\beta = dG\omega$ . $\blacksquare$

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Remarks

RemarkThe Green's operator

The Green's operator $G: \Omega^k \to \Omega^k$ is defined by $G|_{\mathcal{H}^k} = 0$ and $G = \Delta^{-1}$ on $(\mathcal{H}^k)^\perp$ . It satisfies $\Delta G + H = \mathrm{Id} = G\Delta + H$ , where $H$ is the harmonic projection. The Green's operator is a compact, self-adjoint operator on $L^2\Omega^k$ that smoothes by two derivatives.

RemarkManifolds with boundary

For manifolds with boundary, the Hodge decomposition requires boundary conditions. The two natural choices are absolute boundary conditions ( $\iota_\nu \omega|_{\partial M} = 0$ and $\iota_\nu d\omega|_{\partial M} = 0$ ) and relative boundary conditions ( $\omega|_{\partial M} = 0$ and $\delta\omega|_{\partial M} = 0$ ). Each gives a Hodge decomposition, with the harmonic forms corresponding to absolute (resp. relative) cohomology.