Mapping properties of the Laplacian in Sobolev spaces of forms on complete hyperbolic manifolds
Let \((M^n,g)\) be open, complete, \(\Delta=\Delta_p\) the Laplacian, \(\Delta_R= \Delta_{R,p}\) the rough-Bochner Laplacian and denote by \(H^s_p(M)\) the \(L_2\)-Sobolev spaces of \(p\)-forms, based on covariant derivatives. The authors prove the following theorems. Theorem 1. If \(M\) is a complete Riemannian manifold with constant curvature \(-1\), \(\Delta_R\) is a topological isomorphism from \(H_p^{s+2}(M)\) to \(H^s_p (M)\) for all \(s\geq 0\) and \(0<p<n\). Theorem 2. The de Rham Laplacian \(\Delta\) is a topological isomorphism from \(H_p^{s+2} (\mathbb{H}^n)\) to \(H^s_p(\mathbb{H}^n)\), \(0\leq p\leq n\), if and only if \(p\neq n/2\), \(p\neq(n\pm 1)/2\). The point is that for Theorem 1 the authors do not need an assumption concerning the injectivity radius. For other results in this direction the reviewer refers to the paper of \textit{G. Salomonsen} in Result. Math. 39, No.~1--2, 115--130 (2001; Zbl 1056.53029)].
scientific article; zbMATH DE number 2093680