An exact sequence in \(L^2\)-cohomology
The author studies the (reduced) \(L^2\)-cohomology groups of a complete Riemannian manifold \(M\). These can be defined as vector spaces of harmonic and square integrable differential forms. Carron restricts his attention to such \(M\) which fulfill a certain type of Sobolev inequality plus an integral curvature bound. Then in particular the \(L^2\)-cohomology spaces are finite dimensional. The main result is that for such manifolds we get an exact sequence in \(L^2\)-cohomology of the pair \((M,M-D)\), where \(D\) is an open bounded submanifold with smooth boundary. Consequences of this are absolute and relative Gauss-Bonnet theorems for the \(L^2\)-Euler characteristic. The main ingredient in the proof is the construction of a Green operator for the Laplacian (on differential forms) under the assumptions of the main result. The geometric conditions are rather strong. Examples are convex \(n\)-dimensional Riemannian submanifolds of Euclidean space with infinite volume and with \(L^n\)-integrable second fundamental form; or \(n\)-manifolds which are flat at infinity where the volume of balls grows like \(r^n\) (in both cases \(n>4\)).
scientific article; zbMATH DE number 1425165
scientific article