The Dirac operator and conformal compactification (Q2712597)

Let \(g\) be a complete Riemannian metric on the spin manifold \(M\) with Dirac operator \(D_g\). A manifold \(M\) is said to have conformal boundary component if it is diffeomorphic to the interior of a manifold with boundary \(N\) and if there exists a conformal change of metric \(h=g e^{-2\sigma}\) on \(M\) which extends to a smooth metric on \(N\). NEWLINENEWLINENEWLINEThe first theorem of this short note states that under this condition, the \(L^2\)-kernel of \(D_g\) is trivial. In the second theorem, it is shown that the dimension of the \(L^2\)-kernel of \(D_g\) is still finite if \(M\) is conformally equivalent to the difference of a closed spin manifold and some closed subset of finite \((n-2)\)-dimensional Hausdorff measure. Corollaries to both theorems concerning the essential spectrum and the \(L^2\)-index of \(D_g\) are also presented.