Complex hyperbolic manifolds and exotic smooth structures
In earlier work the authors considered the problem of changing the smooth structure on a real hyperbolic manifold while equipping the new structure with a negatively curved Riemannian metric. One technique which was employed was the use of an exotic homotopy sphere flow \(\theta_{n-1}\). In this paper, the problem for compact hyperbolic manifolds is considered. In particular, the authors show that for any positive integer \(m\) of the form \(4n+1\), \(n\in\mathbb{Z}\) (or \(m=4\)) there exist a pair of compact smooth manifolds \(M^{2m}\) and \(N^{2m}\) such that (a) \(M^{2m}\) and \(N^{2m}\) are homeomorphic but not diffeomorphic, (b) \(M^{2m}\) is a complex hyperbolic manifold of complex dimension \(m\), and (c) \(N^{2m}\) supports a negatively curved Riemannian metric. Moreover, given \(\varepsilon>0\), the manifolds can be constructed so that all of the sectional curvatures of \(N^{2m}\) lie in a closed interval \([-4- \varepsilon,-1+\varepsilon]\).
scientific article; zbMATH DE number 599646
scientific article