\(L^2\)-cohomology and Sobolev inequalities
The author proves results about the dimensionality of the \(L^2\)-cohomology spaces of a noncompact variety under certain coercivity conditions and curvature bounds. For example, following the work of Cwickel-Lieb-Rosenblum the author shows: If \((M,g)\) is a complete Riemannian variety satisfying the Sobolev inequality \[ \mu_p(M)\biggl(\int_M |u|^{\frac{2p}{p-2}}(x) dx\biggr)^{1-\frac{2}{p}} \leq \int_M |du|^2(x) dx \] for all smooth, compactly suported \(u\), and if the Riemannian curvature \(R(x)\) belongs to \(L^{p/2}(M)\) where \(p>2\) is fixed, then the reduced \(L^2\)-cohomology spaces \(\mathcal H^k\) are finite dimensional and satisfy \[ \dim \mathcal H^k(M)\leq C_n^kC(p,n)\mu_p(M)^{-p/2}\int_M|R(x)|^{p/2} dx. \] Here \(C_k^n=\dim\Lambda^k \mathbb R^n\) and \(C(p,n)\) is a constant that depends only on \(p,n\) but not \(M\). The author uses different techniques to prove corresponding dimensionality results when \(M\) is embedded into a Euclidean space and one has bounds on its second fundamental form.
\(L^2\)-cohomology spaces of a noncompact variety
scientific article; zbMATH DE number 1336999