Hodge Laplacian and harmonic forms

Let M be a smooth compact connected oriented Riemannian manifold without boundary. The metric and orientation give the Hodge star and the L² inner product on real smooth forms by integrating one form wedged with the star of another. Define δ as the formal L² adjoint of the exterior derivative d. The Hodge Laplacian is $\Delta=d\delta+\delta d$, a degree-preserving second-order elliptic operator. We use the nonnegative sign convention; on functions it equals minus divergence of gradient.

Integration by parts has no boundary term in this setting. Pairing Δα with α gives the sum of the squared norms of dα and δα. Hence a smooth form lies in the Laplacian kernel exactly when it is both closed and coclosed. Pointwise formulas and the adjoint depend on the metric, while the identity d²=0 does not. The sign choice also fixes heat evolution: a form heat equation is written with −Δ on the right so positive eigenvalues decay.

The Hodge theorem gives an orthogonal decomposition of smooth k-forms into harmonic, exact, and coexact parts. The harmonic space is finite dimensional, and every de Rham cohomology class has exactly one harmonic representative. For a closed form, its coexact component vanishes in this decomposition, leaving a harmonic form plus an exact correction. We state the full theorem; elliptic regularity, compactness, and Fredholm theory supply its analytic proof beyond the elementary energy identity.

Uniqueness within a cohomology class is easy to see once a harmonic representative exists. If an exact form dβ is harmonic, its inner product with itself equals the inner product of β with δdβ, which is zero by coclosedness. Thus an exact harmonic form must vanish. The dimension of the harmonic space is therefore the kth Betti number, independent of the metric, even though the actual harmonic representative and its norm can change when the metric changes.

On a compact manifold with boundary, integration by parts produces boundary terms, and harmonic representatives require suitable absolute or relative boundary conditions. Without such a specification, even the implication from Δα=0 to dα=δα=0 can fail. For instance the function x on the Euclidean interval [0,1] has zero second derivative but nonzero differential. The missing boundary term is exactly why the closed-manifold energy argument cannot be copied into that setting.

Noncompact manifolds introduce behavior at infinity and different choices of L² domains. Harmonic functions on the whole Euclidean plane already form an infinite-dimensional family, so finite-dimensional compact Hodge conclusions do not follow merely from a local elliptic equation. On a flat torus, by contrast, constant angular one-forms provide a concrete finite-dimensional harmonic space. Their nonzero periods prevent them from being exact, although the angular coordinate functions themselves are not globally single-valued real functions.