Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?

2011-01-05T06:17:52Z

Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let $\mu$ be the volume measure.

Does anyone know of a reference for the density of $C^\infty(\wedge T^\ast M)$ in $D(d) = \{u \in L^2(\wedge T^\ast M): du \in L^2(\wedge T^\ast M)\}$.

Here, the $L^2$ space is defined as:

$L^2(\wedge T^\ast M) = \{u:M \to \wedge T^\ast M\ |\ \int_{M}\ (u(x),u(x))_x\ d\mu(x) < \infty\}$

where $(\cdot,\cdot)$ is the canonical extension of the metric $g$ to the exterior algebra.

Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold? - MathOverflow

Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?