Hodge Laplacian - Imported Knowlpedia Content

Definition

Let $M$ be an oriented Riemannian manifold. The Hodge Laplacian on differential $k$ -forms is

\Delta=d\delta+\delta d:\Omega^k(M)\longrightarrow\Omega^k(M),

where $d$ is the exterior derivative and $\delta$ is the codifferential, its formal $L^2$ -adjoint. This convention makes $\Delta$ nonnegative: for compactly supported $\alpha$ ,

\langle\Delta\alpha,\alpha\rangle =\|d\alpha\|^2+\|\delta\alpha\|^2.

The operator preserves form degree, is formally self-adjoint, and is elliptic with principal symbol $|\xi|^2\operatorname{id}$ . The same differential expression applies in every degree, including functions; its analytic realization depends on an operator domain and, when present, boundary conditions.

Harmonic forms and cohomology

A form is harmonic when $\Delta\alpha=0$ . On a compact manifold without boundary, the energy identity implies

\Delta\alpha=0 \quad\Longleftrightarrow\quad d\alpha=0\ \text{and}\ \delta\alpha=0.

The Hodge theorem then identifies each de Rham cohomology class with a unique harmonic representative Wells, chapter IV, §2.

On a noncompact manifold or a manifold with boundary, the differential expression is unchanged, but kernel, self-adjointness, and cohomological conclusions depend on domains, completeness, and boundary conditions.

Basic identities

The relations $d^2=0$ and $\delta^2=0$ imply

d\Delta=\Delta d, \qquad \delta\Delta=\Delta\delta.

The Hodge star also intertwines the Laplacian in complementary degrees. These identities let harmonicity pass naturally between closed forms, coclosed forms, and their Hodge duals.

The Weitzenböck formula compares $\Delta$ with the rough Laplacian:

\Delta=\nabla^*\nabla+\mathcal R,

where $\mathcal R$ is a zeroth-order curvature action. On functions, $\mathcal R=0$ , and in Euclidean coordinates the nonnegative convention gives $\Delta f=-\sum_j\partial_j^2f$ .

Conventions

Some analysis texts define the scalar Laplacian with the opposite sign. The formula $d\delta+\delta d$ fixes the sign used here and makes the operator nonnegative in the $L^2$ pairing. The covariant Hodge Laplacian on bundle-valued forms is a related operator with additional coefficient curvature.

References

Raymond O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: chapter IV, §2, harmonic forms, Hodge Laplacians, and Hodge theory.
Shigeyuki Morita, Geometry of Differential Forms, American Mathematical Society, 2001. DOI record. Relevant: the chapter “Laplacian and harmonic forms.”