Harmonic differential form - Imported Knowlpedia Content

Definition

Let $(M,g)$ be an oriented Riemannian manifold, and let $\alpha\in\Omega^k(M)$ be a smooth differential form. The form $\alpha$ is harmonic when

\Delta\alpha=0,

where

\Delta=d\delta+\delta d

is the Hodge Laplacian and $\delta$ is the codifferential determined by $g$ and the orientation. Harmonicity therefore depends on the Riemannian metric. On a compact manifold without boundary, it is equivalent to the pair of first-order conditions $d\alpha=0$ and $\delta\alpha=0$ ; that equivalence need not hold for arbitrary smooth forms on a noncompact manifold de Cataldo, Lemma 2.3.2.

Energy characterization

When $M$ is compact and has no boundary, formal adjointness and Stokes' theorem give

\langle\Delta\alpha,\alpha\rangle_{L^2} =\lVert d\alpha\rVert_{L^2}^2 +\lVert\delta\alpha\rVert_{L^2}^2.

Thus $\Delta\alpha=0$ forces both summands to vanish. Conversely, a closed and coclosed form is harmonic directly from the definition of $\Delta$ . The compactness and boundary hypotheses justify the integrated identity without extra boundary terms or decay assumptions Wells, Chapter IV, §2.

The space of harmonic $k$ -forms is denoted $\mathcal H^k(M,g)=\ker(\Delta:\Omega^k(M)\to\Omega^k(M))$ . On a compact manifold it is finite-dimensional, and the Hodge theorem identifies it with degree- $k$ de Rham cohomology.

Examples and near-misses

On a flat torus, differential forms with constant coefficients are harmonic. In degree zero, harmonic forms are precisely harmonic functions; on a connected compact manifold every such function is constant.

The function $f(x)=x$ on $\mathbb R$ is a decisive noncompact warning: with the Euclidean metric, $\Delta f=0$ , but $df=dx\neq0$ . Hence a harmonic smooth form on a noncompact manifold need not be closed and coclosed under the unrestricted smooth definition. $L^2$ -harmonic forms require a specified self-adjoint realization and are a separate analytic setting.

References

Mark Andrea A. de Cataldo, The Hodge Theory of Projective Manifolds, Imperial College Press, 2007. Author-hosted book PDF. Relevant: §2.3, especially Definition 2.3.1 and Lemma 2.3.2.
Raymond O. Wells, Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Springer DOI record. Relevant: Chapter IV, §2 on harmonic forms and Hodge theory.