Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:44:48Z http://mathoverflow.net/feeds/question/51179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51179/density-of-c-infty-in-the-domain-of-the-exterior-derivative-on-a-noncompact-com Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold? Lashi 2011-01-05T06:17:52Z 2011-01-05T06:17:52Z <p>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.</p> <p>Does anyone know of a reference for the density of <code>$C^\infty(\wedge T^\ast M)$</code> in $D(d) = \{u \in L^2(\wedge T^\ast M): du \in L^2(\wedge T^\ast M)\}$. </p> <p>Here, the $L^2$ space is defined as:</p> <p>$L^2(\wedge T^\ast M) = \{u:M \to \wedge T^\ast M\ |\ \int_{M}\ (u(x),u(x))_x\ d\mu(x) &lt; \infty\}$</p> <p>where $(\cdot,\cdot)$ is the canonical extension of the metric $g$ to the exterior algebra.</p>