The double of a hyperbolic manifold and non-positively curved exotic structures

We give examples of non-compact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three non-equivalent smoothable structures, two of which admit a Riemannian metric of non-positive curvature while the third does not. We also prove that the doubles of non-compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.