Examples of simply-connected Liouville manifolds with positive spectrum

For each n greater than or equal to 3, we present a family of Riemannian metrics g on R(n) such that each Riemannian manifold M(n) = (R(n), g) has positive bottom of the spectrum of Laplacian lambda(1)(M(n)) > 0 and bounded geometry \K\ less than or equal to C but M(n) admits no non-constant bounded harmonic functions. These Riemannian manifolds mentioned above give a negative answer to a problem addressed by Schoen-Yau [18] in dimension n greater than or equal to 3.