LINK TO THE TEXT: A Hodge Theorem for Noncompact Manifolds

ONLINE REFERENCES

Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Annales de l'institut Fourier, 6 (1956), p. 271-355.

Aronszajn N., Krzywicki A., Szarski J., A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., Volume 4, Number 5 (1962) 417-453.

The Hodge Theorem also holds for NONCOMPACT manifolds.

More precisely:

The de Rham cohomology of an ARBITRARY riemannian manifold is computed by the complex of COCLOSED harmonic forms.

The main ingredient used for proving the Hodge Theorem for noncompact manifolds is the article Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Annales de l'institut Fourier, 6 (1956), p. 271-355, by Bernard Malgrange. I'm taking this opportunity to express my admiration for this great mathematician.

The proof results formally from the

LEMMA. Given any (smooth) form A on a riemannian manifold, there are forms B and C satisfying A=LB+C, where L is the laplacian and C is coclosed.

The Lemma follows easily from Malgrange's paper. Again, see Hodge Theorem for Noncompact Manifolds for the details.

For a conjectural AUTOMORPHIC ANALOG, which I call the HARDER-BOREL CONJECTURE, see Section 2 of Potpourri. See also Cohomology of convex cocompact groups and invariant distributions on limit sets by Martin Olbrich.

Here is a statement in the same spirit:

The de Rham cohomology of a real analytic paracompact manifold can be computed with analytic forms.

This results from Proposition 12 page 346 of Malgrange's Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, combined with Theorem 3 page 470 of H. Grauert, "On Levi's problem and the imbedding of real-analytic manifolds", Ann. of Math., 68, 1958, 460-472.